Since the sun climbs above the galactic plane and it even dives below the galactic plane we can't deny that it wouldn't climb upward unless there is something above the galactic plane that pulls it upward as well as something beneath the galactic plane that pulls it downward though it is invisible. Though we don't know how to explore what is it that pulls the sun upward or downward should we not admit that we can't find out what it is instead of saying that nothing exists above or below the galactic plane? Can we agree though there may be a parallel universe above and below the galactic plane, we have no means to explore their existence?
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Your contention is like saying that finding a pendulum on the upswing must mean there is a mysterious force pulling it in that direction - untrue, the only force acting is a restoring force back towards the equilibrium point.
The motion of the Sun in the direction perpendicular to the Galactic plane is perfectly well understood in terms of standard gravitational physics. The mass of the stellar component, which dominates over dark matter at the solar Galactocentric radius, is strongly concentrated towards the plane - that's why it is called the Galactic plane.
Very roughly, you can characterise the density as $\rho_0 \exp(-|z|/h)$, where $z$ is the height above or below the Galactic plane and $h$ is a scale-height of perhaps 200 pc or so.
Applying Gauss' law for gravity, assuming the Galactic disc is uniform on radial scales of kpc, then $$g(z) \simeq -4\pi G\int^{z}_{0} \rho\ dz = -4\pi G \rho_0 h[1-\exp(-|z|/h)]\ .$$
Thus, when a star (e.g. the Sun) has $|z|>0$, there is a restoring acceleration that acts towards the Galactic plane at all times.
Even more roughly, we could approximate $\exp(-|z|/h) \sim 1-|z|/h$, which simplifies the acceleration to $$ g \simeq -4\pi G\rho_0 |z|\ .$$
This is just simple harmonic motion and thus stars oscillate up and down through the Galactic plane, with a period of $(\pi/G\rho_0)^{1/2}$.
The stellar mass density near the Galactic plane is something like $0.1M_\odot$ per cubic parsec. This gives an oscillation period of $\sim 70$ Myr.
Thus perhaps the only mystery is where the Sun gets the kinetic energy to escape from the plane at all? Some of this will be at birth - the 1D velocity dispersion of giant molecular clouds (the birthplace of stars) is a few km/s, compared with the Sun's upward motion of around 7 km/s. Any remaining discrepancy can be explained by "disc heating". This is the process of a gradual increase in the kinetic energies of stars as they circulate around the galaxy, caused by multiple small interactions with other stars, clusters, spiral arms and molecular clouds.
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Nobody says that nothing exists "above" or "below" the galactic plane.
The stars are thickest at the galactic plane and get scattered thinner and thinner with increasing distance from the galactic plane. It is often said that the disc of the galaxy is only 1,000 light years thick, but that is a round figure.
The galactic halo is a spheroidal region of thinly scattered stars and globular star clusters which surrounds the hub and the disc of the galaxy. The mass of the visible stars in our galaxy is a small fraction of the mass of the rather mysterious dark matter in our galaxy, which is more or less evenly distributed throughout the halo.
So about half of the mass of the galaxy is "above" the galactic plane and about half of the mass of the galaxy is "below" the galactic plane.
There are billions of stars in the galactic disc orbiting the center of the galaxy. And as stars orbit around the center of the galaxy, they sometimes pass close to other stars.
When two stars pass close to each other their gravity changes the orbits of both stars slightly.
Here is a link to some past and future close encounters of the Sun with other stars calculated by astronomers. Note that they only cover a few million years and the Sun has existed for 4,600 times a million years.
Each close encounter changes the course of the Sun slightly, and sometimes that change makes the Sun head away from the galactic plane.
Whenever the Sun stars to rise "above" the galactic plane half of the mass of the Galaxy will be pulling it "upwards" and half of the mass of the galaxy will be pulling it "downwards". The farther the sun rises "above" the galactic plane, the smaller the percentage of the galaxy's mass ahead of it becomes, and the larger the percentage of the galaxy's mass behind it becomes.
Eventually the greater mass "below" or "behind" the Sun slows it down to stop its "upwards" motion, and then starts to pull it back "down" again. As the Sun moves back "down" towards the galactic plane it accelerates and builds up enough speed to overshoot the galactic plane, passing through it to the region "below" the galactic plane. As the Sun moves farther "below" the galactic plane the greater mass "above" it instead of "below" it slows it down. Eventually, the Sun's "downward" motion is stopped and it starts to move "upwards' toward the galactic plane again.
And so as the Sun orbits the center of the galaxy within the galactic disc and near the galactic plane, it also oscillates "up" and "down" relative to the galactic plane.
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2$\begingroup$ Can you add in the link you mention to close encounters. Thanks $\endgroup$ Commented Mar 25, 2021 at 13:47
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I hate when everyone else gives a long answer but I give a simple one.
When the sun wanders above or below the galactic plane, what pulls it back down into the galactic plane?
The gravity of all the stars below it in the galactic plane. But it overshoots and wanders out the other side, and gets pulled back in.
Sinusoidal rinse and repeat.
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1$\begingroup$ this is not how orbital inclination works, it exist because the orbit of sun around milky way is not align with the disk of milky way. $\endgroup$ Commented Jun 12 at 16:50
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$\begingroup$ @pm_2_ring Because I'm not rich. I have to do all kinds of things I hate. $\endgroup$ Commented Jun 13 at 10:08
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$\begingroup$ @ProfRob got 16 upvotes for another one of his delightful lectures. I made it 17 because I never knew about Gauss' surface integral law of gravity, only Newton's -- like weightlessness everywhere inside a sphere. That's why it's delightful. I come here for a jolt of understanding I could heretofore only get in college, and it's addictive. But Prof Rob himself said, "there is a restoring acceleration that acts towards the Galactic plane at all times" as a function of z. My post was an incomplete simplification by a mere padawan. But an accurate one. $\endgroup$ Commented Jun 13 at 10:35
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$\begingroup$ @ProfRob The equivalence between Gaussian / Newton, and the equivalence between the divergence of gravity and the divergence in electromagnetism, which I didn't know about, is fascinating. It led me to Dirac's delta function, which I have a question about. But I'm banned for asking here. I don't know why. But I have to find out somewhere else why zero is so special in the delta function, why it isn't just an ordinary surface integral. I have to find out because understanding stuff is like the peak when jumping on a trampoline; I'm floating weightless and I can see forever. $\endgroup$ Commented Jun 13 at 11:53