My question is specifically related to the molecular energies of Kohn-Sham DFT:
Do eigenvalues in KS-DFT mean anything?
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This is a very interesting question and to the best of my knowledge there is not yet a conclusive answer. As the chemistry stackexchange threads linked by @Tyberius already state Kohn-Sham orbital energies and Kohn-Sham orbitals are often used in the chemistry community to interpret computational findings, because empirically this works ok.
But there are problems. For example, there is no guarantee that the lowest-energy Kohn-Sham solution is actually the one in which the lowest-energy orbitals are occupied (i.e. there is no reason for the Aufbau principle to hold for Kohn-Sham). And there are counterexamples . Additionally for neutral and negatively-charged systems, there is no reason for the existence of HOMO-LUMO excitations and in fact already for simple cases like Helium one can prove that they don't exist  (in this case for LDA). This is in contrast to for example the related Hartree-Fock problem, where there always has to be a HOMO-LUMO gap .
With there results in mind, I would say that in general it is far-fetched to directly give KS orbitals a physical meaning. But, people have made efforts to design specific functionals or schemes, which correct on top of the orbital energies based on arguments from the derivative discontinuity. These amount to give results in agreement with ionisation potentials or electron affinites  and therefore could suggest that there is more to it.
So as always in physics, if you check your model applies with a higher-level theory, you might get away with using Kohn-Sham orbital energies and give them a physical interpretation, but there are definitely edge cases where this is not justified.
 Adi Makmal, Stephan Kümmel and Leeor Kronik Phys. Rev. A 83, 062512. DOI 10.1103/PhysRevA.83.062512
 Gero Friesecke, Benedikt Graswald arXiv:1907.00064 (pdf)
 Volker Bach, Elliott H. Lieb, Michael Loss and Jan Philip Solovej Phys. Rev. Lett. 72, 2981. DOI 10.1103/PhysRevLett.72.2981
 Andrew M. Teale, Frank De Proft, and David J. Tozer J. Chem. Phys. 129, 044110 (2008). DOI 10.1063/1.2961035.
A few more papers to add to Michael's answer:
JCP article titled "Increasing the applicability of density functional theory. III. Do consistent Kohn-Sham density functional methods exist?" provides Provide non-traditional view on KS eigenvalues. https://aip.scitation.org/doi/abs/10.1063/1.4755818
If one construct KS potential paying attention of xc potential that is KS eigenvalues of all occupied orbitals should be a good approximation of ionization energies one manage to construct a single correlated particle theory. As a consequence one can observe consistent behaviour in DFT results as shown in this JCP article "Increasing the applicability of density functional theory. IV. Consequences of ionization-potential improved exchange-correlation potentials" https://aip.scitation.org/doi/abs/10.1063/1.4871409
Direct answer: it means the energy of the KS lowest-energetic orbital minimized for a specific Exchange-Correlation functional. If eventually (I don't know how to verify!) the KS lowest-energetic orbital retrieves the system orbital, it means the ground state energy of the system. To my knowledge, we have to be interested in the changes in such energy, not its value.