Where does the idea come from, that possible results of quantum measurement are eigenvalues of the operator corresponding to the observable?
Postulates come in a sense from nowhere, but I will make a case. De Broglie proposed that since light can behave as consisting of particles in the photoelectric effect, it would make sense that all particles could behave like waves. This inspired Schrödinger to replace energy and momentum by the well known operators in the classical energy expression $E=p^2 /2m +V$. It turned out right. The resulting Schrödinger equation has quantized solutions, so called eigenfunctions, and describes (nonrelativistic) physics correctly. And there you are: nature is quantized.
There are two phenomena to distinguish. Quantization is the observed fact that the energy of a system is a sum of quanta of its eigenfrequencies. The eigenfrequencies can be discrete or continuous. This is governed by boundary conditions or, related, discrete symmetries.
If you solve the SE on a discrete basis of functions then the eigenvalue equation becomes a matrix equation.