I was reading up about Nuclear Energy Levels and came up with the aforementioned question. To me, there seems to be no direct connection between the two statements and therefore, I would like to know how does the fact that alpha rays and gamma rays are discrete prove that nuclear energy levels exist.
I think that you may be missing something in the statement that is troubling you. It may be that the statement refers to the emission of alpha particles with different discrete energies from the same radioactive isotope.
Since the parent isotope has a fixed mass, and the daughter isotope also, the alpha particle produced should have a single energy defined by the mass difference in the reaction. And indeed, alpha particles with more than this energy are not seen.
What is seen is a range of discrete alpha particle energies, and a range of discrete gamma ray energies. When these are compared it becomes clear that all the different alpha particle and gamma ray energies can be explained by assuming the parent isotope decays into an excited state of the daughter isotope, followed by the emission of gamma rays as the daughter isotope "drops" down (possibly via intermediate states) to the lowest energy ground state. Here's a graphic example for the decay of Bi-212
The gamma ray spectrum of the daughter isotope is explained by the energy level differences in the excited states.
So in a sense, the "proof" (as commented by @dmckee) is the presentation of a simple scheme that explains all the observed phenomena.
In some cases, the gamma ray emission can be delayed; this product, https://www.pasco.com/prodCatalog/SN/SN-7995_isotope-generator-kit-barium-137-m/ uses a chemically fixed radioactive isotope of cesium. After beta decay, the daughter isotope, Ba-137m, is produces in an excited state (hence the "m"). This isotope can be washed out of the source and its decay by gamma-emission (to ground-state cesium-137) can be measured.
The same effect would be observed in beta-decay, except that the discrete decay energy is masked by being split between the daughter isotope, the beta-particle, and a neutrino...