One of the biggest hurdles I've helped students overcome in electrochemistry, is understanding how the electrons actually move. A lot of textbooks and teachers portray that the "electrons" move through a wire connecting the cathode and anode. Students then get this visual of a single electron making its journey from one end of the wire to the other. However, this is not the case.
When Cu and Zn electrodes are attached by a wire, there is a potential difference. One way to think about this is that Copper actually as a higher affinity for electrons than zinc does, so it "pulls" electrons in its direction. However, a single electron does not travel the length of the wire. Instead, a series of electrons "hop" along the wire from atom to atom in the wire. So two electrons from a Zinc atom in the Zn electrode "jump" to a neighboring Zinc atom, and in the process displace another two electrons which "jump" to another Zinc atom, and so forth. Eventually, 2 electrons "jump" from a zinc atom on the electrode to an atom in the wire, which therefore displaces another 2 electrons which "jump" to another atom in the wire. Electrons continually make these "jumps" across the wire until some electrons "jump" from the wire to the Cu electrode. It is here that they undoubtedly find a Cu2+ anion to combine with and form another atom of Cu on the surface of the Cu electrode.
So the most important thing here is to note that the exact electron(s) which leave a Zinc atom on the Zn electrode, are not the same exact electron(s) which find themselves attaching to Cu2+ anions on the other side. Instead what you have is a bunch of consecutive electron "hops" which looking from the outside appear as a continuous flow of electrons from Zn to Cu.
One way to think about it is having a straw and filling it completely full with beads. Now try to pass a bead trough the straw from one side to the other. When a bead goes in the one side, another bead will pop out of the opposite side. Beads (electrons) were "flowing" through the straw (wire), but the bead that entered the straw is not the same bead that came out of the other side.