I don't understand band theory. How come the density of states at the Fermi level determines if a material is a conductor or not?

Every intro level textbook has some version of this picture: https://en.wikipedia.org/wiki/Electronic_band_structure#/media/File:Band_filling_diagram.svg . Then, there is always some handwaving about how the density of states determines if it is possible to excite electrons, and that whether or not electrons can be excited determines whether or not the material conducts electricity.

I'm stuck on the "possible to excite electrons" implies "material is a conductor" step. I think the reason is that people usually talk about the classical theory of conductivity, i.e. the Drude model, and I can't imagine what the quantum version of the theory of conductivity would look like. How does the wavefunction electron in a metal look compared to one in an insulator?

Also, what about indium tin oxide? How does that work?