It's easier to think about it in terms of energy storage than volts and current. The more energy an amplifier stores, the more current can be drawn from it. It is similar in concept to an electrical outlet for a speaker to "plug into".
An electrical outlet will give as much current as the load demands. If you took a piece of wire and shorted a wall outlet, the wire will burn (as well as the wires in the wall) if there was no circuit breaker. If you use the same wire to short a 9-volt battery, nothing happens. Ohm's law can explain the wall outlet, but not the battery. If the wire has a resistance of a tenth of an ohm, the current drawn from the wall outlet is 120 volts divided by a tenth ohm, or 1200 amps, which is lots of current. However, Ohms law says that same wire will draw 90 amps from the 9-volt battery (9 divided by a tenth). No way that will ever happen, a 9-volt battery will max out to half an amp. That limitation is because the battery does not store a lot of energy. The energy storage for the wall outlet is the transformer on the utility pole, which has tens of thousands times more energy than the battery and therefore can easily supply very high short circuit current.
An amplifier is an energy storage device due to the power supply transformer and filter capacitors. The amplifier power rating at different impedances is a direct measure of the energy storage capability -- the higher the power rating, the more energy is stored, and the more the ability to deliver current to the speakers when the speaker impedance drop demands it .