How can AC current power anything?
@The Photon's answer is quite extensive, the only thing missing is, how electrical energy is now actually transferred. In a simple case where you just have some kind of ohmic load, it is exactly the same as for DC, just with switching polarities.

If you want a picture, imagine a saw: It is pulled through the same block of wood, back and forth. The same sawteeth enable it to remove layer by layer, as there is a force (and power) applied while moving into both directions.

For the electrons, it is quite similar. An alternating voltage keeps pushing them through some load. As they pass through the load, they are moving from a high-voltage node before the load to a low-voltage node after the load, giving off the energy difference between the first and second state.

Then the AC polarity is inverted and again, they are on a high-voltage node, passing through the load, to a low-voltage node. Again, their previous state had more energy, thus the energy is transferred into the load.