On mathematics behind molecular orbital energies
An internal or interior eigenvalue problem seeks a solution somewhere in the middle of the spectrum of eigenvalues as opposed to trying to find the smallest or largest eigenvalue. Its generally trickier to solve for a particular eigenvalue than it is to solve for the smallest/largest one. The HOMO and LUMO are necessarily somewhere in the middle of the eigenspectrum.
The Kohn-Sham Hamiltonian is the orbital-DFT approximation of the molecular Hamiltonian. The Hamiltonian describes the interactions within the molecule and allows one to solve for the molecular wavefunction through the Schrödinger equation. Most of electronic structure is focused on approximations of the Hamiltonian or the form of the wave-function that make solving this problem tractable. There is a lot more that can be said about Hamiltonians than would ever fit in an answer here, so you should try to find good textbooks and papers that discuss them.