Introductory mathematics is not done with formal definitions. This is because most people at the corresponding age (kids) don't know what the words "formal" or "definition" actually means. This only come to play much later.
You start teaching kids mathematics by showing them how to count things up to ten using their fingers. Then, you teach addition with little simple problems like "John had five apples. Mary gave him two more. How many apples does John have now?" - You teach by making associations of numbers and math with phenomena that are easily recognizable and verifiable in their world and day-to-day activities.
Only after they are already somewhat used to counting numbers and doing addition, subtraction, multiplication and division and they already got a few lessons on sets of oranges, sets of cats, sets of some small numbers and union and intersection of those is that the concept of the set of natural numbers ($\mathbb{N}$) are introduced to them.
Some time further, you introduce the concept of fractional numbers and negative numbers. And again, this is associated with things easily recognizable in the mundane world like "two and a half pizzas" or "a half orange" or "John have five dollars and promised to give seven to Jane, so he is lacking two". So far, no one gave them a formal definition of those things, these are only show up later on when the kids are (or should be) firmly familiar with the concept.
The (in)formal definition of $\mathbb{Z}$ only comes in after the kids did played a bit with negative numbers. Only after the kids are already familiar with fractions is that you explain that "every fraction is representable with a ratio between two integers numbers" and you "call all those numbers that can be represented in this way as the set of rational numbers denoted as $\mathbb{Q}$". Gosh, that was an informal definition, but it is sufficient and more than enough to be understood by kids. No kid would easily and quickly understand something like $\mathbb{Q} = \{\frac{p}{q} | p \in \mathbb{Z} \land q \in \mathbb{Z}^*\}$ because that notation and the rules governing them are alien-like for most of them even if they were already presented and used to all of the involved elements.
Some time later, when teaching square roots and geometry, things like $\sqrt{2}$ and $\sqrt{3}$ show up. You quickly tell them that these aren't rational numbers and that there is no way to represent them as a ratio of two integers. Again, that is an informal definition, but is enough. You might even give a further step by telling them that any square root of a prime number is irrational because that if that was not the case, by finding a ratio of two integer numbers that squared give the supposed-to-be prime number would mean that it would be composite (i.e. a proof by contradiction, but still informal). Then, you present the symbol $\mathbb{I}$ to represent them. Also, soon $\pi$ will also show up to join the irrational numbers group.
Finally, you present the real numbers just as "the union of rational and irrational numbers". This is a very simple and informal, although very precise and sufficient definition.
So, the answer is that people can learn about the real numbers without formal definitions by simply working to them with informal definitions and correlations with real-world mundane concepts and then by building up higher-level concepts on top of lower-level concepts. Formal rigorous definitions are then usable only for people that are already skilled enough in math to be able to make sense of them.