I understand that when d-orbitals split some electrons can occupy the higher orbitals by absorbing frequencies of light, but do these electrons then fall back to their original ground states? If so, why? Why don't they remain in a permanently excited state?
I've heard someone mention that the electrons do indeed fall back to ground state but, you don't see that part of the emission spectra since the energy is dispersed everywhere and not just towards your eyes. This is why the colour comes from the absorbance spectra.
The reason is does not stay in the excited state is simply because it is unstable. It will gain energy and then release it as mentioned above, then repeat.