In the ground state, there is no net current because the group velocity of electrons with opposite wavevectors exactly compensated each other.
When the valence band is full, you cannot make an infinitesimal change of the energy (this is the excitation gap). So the linear response to an electric voltage is zero, and the system is an insulator (or semiconductor).
When the valence band is not full, you can slightly deplete the occupied states near the Fermi surface on one side (say with negative k and negative velocity) and fill empty states near the Fermi energy on the other side (with positive k and positive velocity), resulting in a net current. One has a non-zero response to an inifinitesimal applied voltage, so the system is a conductor.