How did the scientists explain the relationship between the colors observed and the structure of the atom?

Every one color match up to light of a certain wavelength produced by the atom when an electron marks a changeover among permissible orbitals. Each color signifies a jump of the electron with resultant emission of photons of certain energy conforming to the vertical colored stripes. The scientist can assess these energies by means of einstein equation for the energy of a photon which is E = h. v where v is the frequency of the emitted photon that can be assessed by means of the wavelength and h is a constant. Then it can recreate the jumps and accordingly figure out the orbitals and perhaps the map of the structure of the atom. 

This post might have the info you need: