In many materials, the outer electrons are not confined to an orbital around a single atom. The orbitals spread out, extending across many atoms. Many electrons overlap. The Pauli exclusion principal says overlapping electrons cannot all have the same state. The orbitals have a range of slightly different energies and momenta. This forms a band.
Sometimes the electrons fill all available states in the band. These materials are insulators. For every electron with a momentum to the right, there is another to the left. There is no net transport of charge.
Sometimes the band is half full. This is a a conduction band. There are empty states at energies just above the most energetic electron. This material is a metal. Metals are good conductors because an electric field can easily excite an electron to a state with a momentum in the direction of the field.
To make a mirror, metal is polished to a smooth surface. Usually the surface is a plane. This defines the boundary condition for the electrons.
Typically a mirror is used at a distance from the light source. The light can be reasonably approximated by a plane wave. The approximation is not perfect, and the light is often not monochromatic.
Light is comprised of photons. If light is a monochromatic plane wave, the wave function of each photon is a plane wave. If not, the wave function is a superposition of monochromatic plane waves with various directions and frequencies. We can understand what happens by considering a monochromatic plane wave photon.
When a photon strikes a mirror, it will be absorbed by one electron. That electron will be excited, gaining energy and momentum. Later, it will drop back to the previous state, emitting a photon.
Neither the photon nor the electron have a well defined position. Both can be decomposed into states with well defined energies and momenta. The momentum of the photon can be resolved into components parallel and perpendicular to the plane of the mirror.
The momentum of the emitted photon will have the same parallel component, but a perpendicular component in the opposite direction. I can't give you a quantum mechanical reason for that. As one of the comments indicated, the theory of how light and electrons interact is quantum chromodynamics. Perhaps someone familiar with it can add to this?