I know that in the Kronig Penney model there are values of the energy $E$ for which solutions to the Schrodinger equation don't exist. I understand that these forbidden values of $E$ form the band gaps. How, intuitively, does the crystal structure lead to the existence of forbidden bands?
Diffraction of electron waves inside a periodic crystal structure, i guess? Superposition of wave functions. That is the intuitive way. One more thing, you can get this with perturbation theory too.Or with tight binding approximation. Anyways for electrons that satisfy special condition there will be constructive interference and they will interact strongly with the lattice. This happens exactly on the edge of brillouin zone...and you get from this linear combinations of incoming and reflected waves which then give you possible forms of wave functions. In one dimension you get two forms id est two combinations. When you write down the probability distribution and plot it against the lattice, you get that for the first linear combination electrons are piling up between ions and for the second near the atoms, so you get two distinct energies because of interaction. Where does the wave vector come in play? Through the Bragg condition where we see that this happens for special values of k-vector...and this is definition of Brillouin zone.
The essential physics, to me, of the energy gap is interference/diffraction of the electron wave. In a very good crystal, like GaAs or graphene, an electron can travel many µm, traversing tens of thousands of atoms. If you consider this classically, it is astounding, as a classical particle would simply collide with the nuclei. However, the reality is that, because of the wave nature of the electron, the wave spreads out from being localized to a single nuclei. We say that the electron is delocalized, and can carry current. Of course, this is for electrons of the correct energy, falling in one of the energy bands. In the perfect crystal, no electron of forbidden energy can exist in its bulk. Real imperfect crystals are even more interesting, but that's a discussion that can lead to a career!