Why do electrons have fixed energy levels? I understand that electrons do not orbit the nucleus, instead they have a higher probability to be found at some specific regions.
But what makes they appear more frequently in the orbital regions? There are equations (like Schrödinger) that are able to describe this wave function, but what causes it?
 A: Nobody knows the answer. All we know is that quantum mechanics is in perfect agreement with experiment. All you can do is to critically investigate any intuitive concepts that you may have that are incompatible with quantum mechanics.
A: I believe that when Richard Feynman was asked what the Schrödinger  equation (SE) was, he replied (paraphrasing from memory): 'it's something that happened in Schrödinger's mind!'
In truth we don't know why the SE works, only that it does tremendously well!

I understand that electrons do not orbit the nucleus, instead they
have a higher probability to be found at some specific regions.

The SE is a second order, linear partial differential equation and quite similar to other important equations like the (Classical) wave equation, diffusion equation and Fourier heat equation.
Like the SE, when these equations act on a bounded domain they become eigenvalue/eigenfunction problems. In Quantum Mechanics we call this quantisation.
In the case of the SE, the bounded wave functions (solutions to the SE) combined with the Born rule this yields the probability density distributions which tend to be wave-like.
A: The energy levels of the Hydrogen atom have specific energy levels because the solution of the S-equation says so. Other than this no one knows why things are described by the S-eqn. It just the way things are. The S-eqn provides a mathematical explanation of the way things work. It has been checked against known results and it agrees perfectly so there is little reason to doubt it.
