Imagine a hollow cylindrical tube, several yards in length, and filled with spheres lined in its inside. (A diagram would help here, but the idea is that the spheres completely fill the tube, and can move to either side).
Now, picture this 1) If you insert an additional ball in one of the sides of the tube, almost instantaneously, another will drop at the other end of the tube; 2) If you continue to insert additional balls on your side, a stream of balls will be dropping out of the other side, but the actual speed of the individual balls is small. Actually you will recognize that the first ball you inserted will only come out on the other side after all the original balls in the tube were replaced.
This is what happens with electricity. When you drive electrons into a wire, they push the electrons already there in the metal, which push further into it, and further into it, and further into it. The result is that the electrons entering the conductor create a signal that propagates very fast to the electrons on the other side, but the electrons themselves are not making the trip immediately. Physically, they have two speeds: random thermal motion which is fast but in position averages out to zero, and the drft velocity (about 1 millimeter / second) that carries them along the conductor.
Notice that, for an electron that has entered, to exit on the other side, you'd need to replenish all electrons initially in it, so to speak, and that number is trully very high.