This is a major subject, based on quantum mechanics. We are using a pretty simplified picture, but one that has the important features for understanding how solid state electronic devices. In quantum mechanics, electrons in atoms are only allowed in certain energy levels (solutions to the Schrödinger equation). In crystalline solids, like silicon or germanium, the solutions to the Schrödinger equation need to take into account interactions, and the resulting solutions correspond to bands of allowed energies. There tends to be a band of allowed energies available to the ``valence'' electrons outside the last closed shell, corresponding to electrons localized around the nuclei. There is then often a gap in energy, representing energies that electrons are not allowed to have, and then another band of allowed energies, corresponding to delocalized electrons. When electrons are in this upper band, the ``conduction band'', they are allowed to move around and conduct current.
So, only when electrons are promoted into the band (by thermal energy, a battery, photons, etc.) can they carry current. Materials for which there is little or no band gap are good conductors.