This is, I believe, a shortcoming of a lot of poor use of language and also of unfortunate pop-sci explanations involved here.
Quantum mechanics does not say that things "need an observer" to "exist", any more then that classical mechanics does. One can say that it is philosophically debatable, of course, whether things "exist" when we aren't looking, but what I'd say is that quantum mechanics at least does not shed any more or less light on this than it already had.
You have no doubt heard things like "well the object doesn't 'exist' or 'doesn't have properties'" prior to being "measured" or "observed". This is not right. A better interpretation is that it has fuzzy properties, and in fact it has them at all times. Even when a measurement or observation is made, at best all this does is clarify one property at the expense of another, which is the tradeoff encapsulated in Heisenberg's uncertainty principle which, by the way, is perhaps better translated from the German as the "fuzziness principle", or even "blurriness principle", where the term translated as "fuzzy" or "blurry" here is the same word that would be used to describe a blurry photograph, i.e. when the lense on the camera is defocused and a photograph taken.
What "fuzziness" means here is that there is a restriction on the level of information which defines the properties of the particle. In Newtonian mechanics, properties of a particle such as its position are defined "with an infinite amount of information": the variable $x$ representing position, at least mathematically, is an infinite-precision real number. It gives us perfect information, singling out one point in space with absolute certainty. Moreover, in general it would require an infinite amount of paper to write it all down.
The notion of "limiting" the amount of information is what leads us to use probability distributions. Probability distributions are, mathematically, how we represent a situation where information is missing and, indeed, one should not find this concept too unfamiliar. If someone tells you they're "only 85% sure" about something, it means the information they have about it isn't really as good as someone who can be 100% sure or 0% sure (i.e. 100% sure of its falsity). The overall degree of privation of information for a particular distribution can be quantified by the Shannon entropy, which for a finite or countable set of outcomes with probabilities $P_i$ is defined by
$$H := -\sum_i P_i \lg(P_i)$$
where $\lg$ is the base-two, or binary, logarithm, which we customarily use if we want to measure the entropy in bits - if we want to use a different unit, we should use a different base (if the base is $e$, you are measuring in "nats", and if the base is 10, in "hartleys"). The higher $H$ is, the more information we are missing as a result of nontriviality in the probability distribution. The "shape" of the probability distribution represents in what way we are missing information, and the wide range of possible variation in that shape corresponds to great variety in the possible ways we could be lacking it. We can lack information about a situation in many ways - consider, and indeed this will be a bit close to what we're talking about since it concerns location, e.g. a pet cat (not originally inspired by a more famous one but upon second look, rather apropos for this discussion, imo) left at home while we go to run an errand (so at least, thankfully, we are not some sicko with a sadist complex who murders little animals pointlessly with painful gassings simply to sate hir own curiosity). We will expect the cat to wander, and thus when we come back, we will not know necessarily in what room of the house we will find it when we go inside again. Alternatively, we may know that - if it's the right kind of cat - it will stay in one place for the most part, but that there may be a friend we entrusted with access to the house coming over during our outing, which will cause it to move, but we don't know that for sure. We will end up then with two different distributions of probability for the location of the cat when we are going to open the door - two different ways in which we are missing information. In both cases, one way we can describe them mathematically is as a probability distribution $P(\lambda, \phi)$ of the cat's geographic position on the Earth (measured to suitably fine resolution), here given as latitude $\phi$ and longitude $\lambda$.
As a note, in physics, for a continuous variable like the position of a particle, this sum won't do, because there are an uncountably infinite number of possible points. Instead, we must take an analogous integral over the continuous probability distribution:
$$H := -\int_S P(x)\ \lg(P(x))\ dx$$
Unlike the case of the discrete-space probability distribution, thanks to the scale invariance of the continuum, this entropy effectively has within it a "reference level": namely, when the entropy $H$ is zero, it generally doesn't correspond to having complete information. In fact, the entropy can go all the way down to negative infinity. Rather, roughly you can take a zero of $H$ to mean that we know the position to within the equivalent information of knowing it to one unit of the scale $x$, which may be set variably to be meters, millimeters, nanometers, picometers, etc. and $H$ will change accordingly. A value of $H$ below 0 effectively corresponds to how much information we have in refining the position to below the level of our measuring unit scale, and above, that we can't even narrow it down to that. The negative of $H$ can be considered the degree of information presence, $I$.
So where does the "observer" enter in to all this? Well, the reason for that is that if we're going to be talking about information, we also need an information bearer, and while of course physical systems are information bearers, in the conceptual background of information and probability-as-information theory, a probability distribution is taken as representing the (incomplete) knowledge, or information possessed by, one information-processing system about some external entity to itself. The key part is that this information-processing system need not be a human: indeed, the theory behind this - information theory - is used all the time in describing, say, communications between computers even without any humans interacting with them in the process thereof. Ultimately, of course, since our theories are supposed to be used by us to explain the world we see around us, the "ultimate" one to be informed is typically a human, but there is no reason that we cannot narrate a story about the Universe without humans, or narrate a story from a non-human's point of view or perhaps better and more humbly, "our imagination of if a non-human were the expositor" (might look into, say, Karen Barad's theories if one wants some philosophical fluff grounding to pad this out although her writing style is kinda difficult which imo is a bit sad and I might not be making precise this statement though I don't feel like digging this out. Feminist philosophy actually seems to do a better job at handling this, imo, than maybe the kind of philosophies considered "kosher" here but then again I don't give two craps what's kosher, just what actually illuminates.) or even simply the attribution of viewpoint and better, agency, to such.
Indeed given this, I'd suggest a good replacement for the term "observer" here is "agent". In quantum mechanics, putting this together, we have an agent - basically any system that can store, retrieve, and process information and through interaction with the outside world, acquire information thereabout. To avoid taking the various terms in quantum mechanics too literally like that the "wave function is a physical object" or that "probabilities disappear from some regions of space" as though they were a 'substance' or any of a number of other such things, we should add the qualification that what it constitutes is a mathematical model for the knowledge/information possessed by that agent and what information it can deduce therefrom, and how new information can be acquired from the outside world, and it is from the perspective of such an agent - whether human or non-human - that the theory describes the world. The probabilities, etc. are all just language that we use to describe such. The agent may encode its actual information store very differently.
I'd also want to point out that even in classical mechanics we can't be entirely devoid of an "observer" because ultimately we need a reference frame, and yet for some reason this does not seem to cause as many quibbles, but I suspect that is because quantum theory is painted as being far more magic and "indecipherable" than I think it really deserves to be.
In that model, the information the agent has about a specific physical parameter of a system, say, the position $\mathbf{x}$, is modeled by a wave function $\psi(\mathbf{x})$. More generally, it can be modeled by a Hilbert space vector $|\psi\rangle$ which can be projected to get the information for many different parameters. This is just standard quantum theory, more or less. The squared magnitude of the function's values gives the probability distribution representing the information possessed by the agent about where the particle is, while the values themselves contain an interesting phase factor that, while not corresponding to anything directly observable, is indispensible in describing dynamics as it gives rise to both interference patterns and also the "quantization" from which "quantum mechanics" takes it name.
Now you may object that $\psi(\mathbf{x})$ also takes "an infinite amount of information" to write down, and perhaps even more, being now a function in space, and thus how can we say the agent "has limited information"? That's a valid point, but it's a subtle one: keep in mind the earlier discussion. The information describing $\psi(\mathbf{x})$ is information describing in what way our agent does and does not have information. The amount of information the agent has about the position is the (negative) of entropy.
But as a scientific theory, of course quantum mechanics should allows us to predict observations not yet made, and thus we have to add some more elements now to give us that ability and to that we say that in addition to the basic fact of the wave function, there are also two things that the theory's user (not conceptually and not necessarily the "agent" in the theory whom is ascribed $\psi$ as a model of its knowledge) can do with it:
There is an operation which we may call EVOLVE, which basically tells us, "given a wave function $\psi(\mathbf{x})$ describing what information our agent has at a present time $t$, what is the best information that the agent can possibly have at a future time $t + \Delta t$?" This is basically the Schrodinger equation.
There is another operation which, and this is what causes all the hoopla, we may call, to get around that, QUERY. The operation QUERY basically, as the name suggests, consists of "asking a question" of the external system, e.g. "Is the particle located at $50-100\ \mathrm{pm}$ from the atomic nucleus?" or "What is the current energy level of the particle?" or any of an (infinite) number of other such possible questions. In this operation the agent acts upon the external system so as to retrieve that information. When the information is received by the agent, it updates the information it has with the new information - which means the probability distribution $\psi(\mathbf{x})$ is changed. The rules for changing $\psi(\mathbf{x})$ in the theory depend on the precise nature of the query in question and how we are modeling the querying process and mechanism.
Now that last part should be highly emphasized. In this second operation, the change here does not represent anything physical in the system: it is simply an update of the knowledge that our agent has. If a broad probability distribution in space becomes a narrow one after this, that doesn't correspond to some physical "instantaneous disappearance" of matter or some kind of "energy" or something else from some parts of space. It just means the new information eliminates those regions and in fact, the actual effect is to increase the information content - lower the entropy $H$.
That said, however, there is a real physical difference at work here and, in fact, it has nothing much to do directly with the fact we are modeling in terms of agents and their acquisition of knowledge. In fact, you can model classical mechanics using probability distributions and incompletely-informed agents in just the same way: that's how one would describe the original cat scenario I just gave earlier as the cat is quite far and away from the realm of quantum mechanics!
Instead, the actual differences, and what is the "real" physical content of quantum mechanics as a theory, is the uncertainty relations or, in more general terms, the non-commuting nature of the operators representing certain physical parameters, namely those that are Hamiltonian conjugates in classical mechanics. What these do is they result in situations where that if the agent becomes more informed about one parameter, it then becomes at the same time less informed about others, at least when the information requested in a QUERY operation is sufficiently strong, and "sufficiently strong" is encoded by the constant of non-commutativity: $\hbar$, or Planck's reduced constant.
And it turns out this necessitates also that any suitably informative query will necessarily have a physical effect upon the system queried and, moreover, that also if we want to reduce the physical effect to zero, we can only do so at the cost of also providing zero information to the agent. This is the real content of the "observer effect" as being a profundity of quantum mechanics. In fact, observer effects - the general idea where the process of observation changes that which is observed - are far from limited only to, or some mysterious component of, quantum mechanics. They are found all throughout many areas of science. There is an observer effect in computer programming, especially when doing timings or memory debuggings. Sociology and psychology are notorious for them. Heck, even in classical mechanics, technically there is an observer effect, it's just left out of the presentations in question: to find out where something is located, I have to bump it with something, and you may also have heard this as an "explanation" of the Heisenberg principle in quantum mechanics. However, in classical mechanics, at least when you have such a "bump", you can make the "bumping" object arbitrarily light and non-intrusive as to the energy and momentum that it imparts. In fact, you can even do that in quantum mechanics, too! What the quantum laws and the true content of HUP is, is that when you do this, you also get *arbitrarily little information* about what you're observing.
And going a step further, this reveals that what the theory is really saying about the Universe is ultimately that it contains an information content limit, in the same way that Einstein's relativity theory tells you that the Universe contains an information propagation limit. The limit on information content is set by the physical constant $\hbar$, just as the limit on information propagation is set by the physical constant $c$ (that $\hbar$ doesn't directly contain units of information is more owed to the subtleties of measuring such, than of it not actually being such.). The introduction of agents, probabilities, etc. is all just the necessities of our language to describe that situation and describe it in the ways that are convoluted enough to make an accurate (very accurate!) theory about the Universe. How the information is "really" stored in the Universe is something we can't know, but this, again, is a philosophical problem that isn't strictly speaking limited only to quantum mechanics. We could have just as well asked the same of classical mechanics, were our universe classical, with "how does it store infinite-precision reals?" and so forth.
Now finally, to wrap up with "how can effects occur without an observer?" Just because the description requires an observer - or here, an "agent" - doesn't mean then we are unable to offer a narrative of the Universe before us humans existed. Already, classical mechanics even since the time of Galileo with his famous thought experiment involving the ship on the perfect sea, had effectively done away with the notion of a completely observer-independent description of reality, yet it seems that in discussions of quantum theory this often gets forgotten about or simply ignored, by doing away with the notions of absolute motion and absolute rest. And indeed, modern developments in physics only serve to push things more and more in a relative direction, and quantum mechanics is in fact part of this trend, not something totally different therefrom. The big step it makes over classical mechanics is really the aforementioned upgrading of the passive "observer" to an active "agent" whose "observations" actually not only affect the Universe but moreover must have such effects to at least some degree if they are to be informative to it. This upgrading is effectively a necessity that follows on from the need to talk of limits on information content, which require measures of information be introduced and then probabilities, and by bringing information in in a central way, the transaction of information between the external object and observing agent becomes important.
How, then, do we make the description? Simple - just as if we're going to make a classical model of the Universe wherein we pick some reference frame, we likewise simply pick by fiat a fictitious "agent" who tells the story from the beginning even if one could not exist physically. The only trick is, however, that due to the agent's interactions it will of necessity introduce some possible deviation in the evolution and thus could be considered to at least in theory throw the predictions of the theory given its fictitious nature: however, we can remedy that by simply considering the level at which it extracts information to be suitably coarse, and/or its queries to be very infrequent.
(One may also wonder what role "decoherence" has in all this and how it supposedly "obviates" the need for operation QUERY above. Decoherence is really just an effect describing how that the wave function behaves when we are considering a wave-function description of another agent. The end wave function ends up being one in which the agent is superposed between multiple outcomes of its query, while in each such outcome the individual queried object has been "reduced" to one outcome. To understand the superposition, remember that since every wave function must be attributed to an agent, when we do this description we are implicitly presuming a second agent is in play. The superposition means here that this second agent doesn't know - due to the nondeterminism, which is the manifestation of the relative lack of information in the Universe - how the first agent has been updated by its query. It will be resolved when the second agent asks - if it's a human, likely literally asking, if it's a computer, then by sending it a request for data, etc. just to avoid anthrocentrism - the first for its result. The real, physical "observer effect", where the query/observation changes the reality, and which is erroneously advanced to "explain" the HUP in the naive "ball bounce off" idea, is embodied in the fact that in this evolution the queried system also undergoes a change to the given outcome during the decoherence event. Confusion results from not cleanly separating conceptually the subjective knowledge-update in operation QUERY from the similar-looking physical "system collapse" in the decoherence process which actually could be interpreted as increasing the information content of the queried system. Similar-looking does not imply identity! One is a real physical effect, the other happens within the agent [according to our model thereof]. The two are not completely unrelated, either - when one agent goes to quiz the other on what it saw, the result of that quiz better be something sensible like "I saw a particle at X" or "I saw a yes/no" and not some weirdly surreal mosh of various options. Decoherence, instead, is thus simply the theory telling us that it is internally self-consistent - hooray!)
