I don't know of any easy way to explain this, it is not the movement of electrons in a pure transporting sense. If it is from moving of electrons, it will take a long time to go from one end of the wire to the other end. Drift velocity is very slow, the better the conductor, the slower electrons move.
If you apply a voltage at one end of the wire, you'll see the voltage at the other end of the wire traveling at almost light speed from the applied end to the measuring end. It is the EM wave that travel. Remember to every source, there must be a return path of electricity. The forward wire and the return wire form a wave guide that EM wave propagate in between. The voltage and current is the consequence of the boundary condition of the EM wave...
But of cause I can just tell you that when you apply a voltage at one end, the current travel from one end of the wire to the other end. That is what most people thing about and how most of the books' explanation. For the most part, it is good enough and they developed most of the formula using this concept. Just remember my first paragraph, if it is really electron movement, you can inject an electron at one end of the short cable, you'll have time to get a cup of coffee, then come back and wait for that electron to come out from the other end.