Consider a train. When the engine starts moving, it pulls/pushes all of the cars along. For electrons, the push goes at the speed of light - when an electron at one end of the train moves, the one at the other end feels it at a time c/L later. But the train itself moves very slow.
But current is the number of electrons moving past a point per unit time. The key here is that there is more than one track - copper atoms, for example, are spaced only a couple of Angstroms apart, and each can potentially contribute a conduction electron to current flow. Here's a quick estimate: a 1 mm thick wire is several million atoms across, or 1012 or so atoms in area, giving 1021 atoms per meter. An amp is 6.25 x 1018 electrons per second, so these 1021 electrons only have to move 6.25 mm/sec to give you that current.
(The thermal speed of the electrons at room temperature is much higher than that, but that's random and doesn't contribute to net current flow)