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.