how can the existence of spectra help to prove that energy levels in atoms exist?

Answer 1

This is a semantics question because spectra, in this context, means the same thing as energy levels. Therefore, if find spectra you have found energy levels.

Answer 2

IIRC, take for example a (thin) gas heated to the point that it emits light. It will only emit protons of specific frequencies. The frequency of the photons correspond to the energy levels that exist in the atoms. Specifically, photons of some specific frequencies are emitted whenever an electron moves from a higher energy level to a lower energy level.