Under normal operating conditions, Zener diodes do not undergo an avalanche breakdown. An avalanche breakdown is (usually) disastrous, irreversible and generally undesirable.
The Zener diode operates based on quantum tunnelling. To describe it simply, a high reverse bias voltage would increase the potential energy of the valence band electrons on the P side (since it would be connected to the negative terminal). Think of it as the electrons on the P side being strongly repelled.
At a certain point, the potential energy of these P side valence band electrons becomes so high that they actually possess a greater energy than some of the unoccupied energy states in the conduction band on the N side. In other words, it becomes energetically favourable for them to be in the conduction band on the N side.
Given a sufficiently narrow depletion region, electrons from the valence band on the P side can spontaneously move to the conduction band on the N side via quantum tunnelling. This constitutes a current. If it helps in visualisation, you can see this as holes being injected into the P side valence band and electrons being injected into the N side conduction band.
The reason the voltage drop is maintained is that the potential energy of the P side valence band electrons must be maintained above the potential energy of the N side conduction band electrons in order for the flow of current across the junction to remain energetically favourable.
To put it as succinctly as possible, the process involves creating a junction voltage so high that electrons on the P side break free from their covalent bonds and move to the N side, creating a current. The voltage remains because it is required to maintain a constant flow of current by making it energetically favourable for electrons to move from the P side to the N side.
NOTE: The potential energies I talk about are with regards to electrons, voltages and potentials in conventional circuits are described with regards to positive charges. That's why I can describe the potential energy of electrons near the negative/ground terminal of a battery as higher than those near the positive terminal