Why do electrons flow through a wire?

The electrons do literally move, both in AC and DC. However, the movement of electrons and the transfer of energy do not occur at the same speed. The key is that there are already electrons filling up the wire all along its length. A common analogy for electrical current in a circuit is the flow of water through pipes.

Wires (the metal inside of them) always have electrons inside of them (the atoms). It's only once energy from a source makes contact, it pushes the electrons through through the wire in which ever direction it is connected.