Let's get one thing clear here. Charge does not, Not, NOT move from one plate to the other (i.e., it does not jump across the insulation medium that separates the two plates). :happysad: Are we on board with that statement? :happyyes: I will assume so, and I will proceed.
The simplest explanation of how a capacitor works, and in particular how it is able to allow current to flow across the apparently infinite gap betweeen the plates, starts with recalling that opposite charges attract, and that like charges repel. As negative charge starts to accumulate on one plate, consider the viewpoint of an electron on the opposite plate. It is negative, and there is a growing negative charge across the gap. The electron on the second plate will therefore feel a push, simply because it possesses the same type of charge as the charge on the first plate. So the electron on the second plate begins to move. Now, from the viewpoint of a person watching the wire that is attached to the second plate, it appears that negative charge is in motion within that wire, and that is the definition of current.
If you connect a capacitor to a battery (i.e., a DC source), then negative charge will start to build up on one plate, and electrons will start to leave the other plate. This goes on for a while, but has to come to an end at some point, since there is a limited amount of electrons on the second plate, and since there is no process to bring any electrons back to the second plate. Thus, current will flow for a while, then current will stop flowing. In the "steady state" or "equilibrium" condition, a capacitor will not carry current from a DC source.
If you connect that same capacitor to an AC source instead, then half the time the source is causing negative charge to build up on one plate, and the other half the time it is causing negative charge to build up on the other plate. Unlike what happened in the DC scenario, the reversal of the voltage source is the process that brings electrons back to the second plate, and on the other half cycle it brings electrons back to the first plate. The "steady state" condition is therefore a continuous flow of current through the capacitor. That current follows the same up/down/up/down pattern as the AC source. But it is reacting to the source, so the current does not reach its peak at the same time the voltage of the AC source reaches its peak.