The Chem Libre talks about a different metal, but staying with Zn and Cu there are the Wikipedia articles
https://en.wikipedia.org/wiki/Galvanic_cell https://en.wikipedia.org/wiki/Daniell_cell
I don't see what is so difficult about the concept of an electron being withdrawn from one metal and donated to another in a different place allowed by conduction instead of an adjacent place. Instead I'd even think in a different place is easier to imagine what actually happens, as locally you are dealing with a more homogeneous situation.
You seem to be asking about physically what in particular is driving it. That's a secondary question compared to the main factor to grasp – that the identical same factors are driving the straightforward chemical reaction as drive the one with the electric current. Whatever these factors are.
Crudely the setup Zn(s) + Cu
2+(aq) has more energy than Zn
2+(aq) + Cu(s), which will make the spontaneous tendency for the first pair to be transformed into the second, transferring electrons, one way or another. So the energy of four components come into it.The first pair is different from the other pair so there has to be an energy difference; which way round is another matter.
It turns out, the articles tell us, there is not much difference in the energies of the hydrated cations, so this is not an important factor.That however is not a fact you could be expected to know. More important, we are told, is how the metal atoms fit in the solid metal structures. The Zn atoms fit less comfortably into their solid structure than the Cu atoms into theirs. The reason for this we are told is that the electrons in transition metal solids can form what they call d-orbital bonding, which seems something like the metal-ligand bonds that transition elements also form. To which, depending what level you have reached in chemistry, you might be saying "if you say so". Sort of thing that transition elements do. But nothing the inventors of the original cells could have predicted. I don't know that even the direction of the reaction is predictable, maybe someone at home more than me with these things could point out some systematic tendencies from which it is.
Maybe instead of messy chemistry some clean physics would illuminate you?

An electrochemical cell does not need two different metals – both of the electrodes could be Zn. Then if they are in solutions of different Zn
2+ concentrations there will be a calculable electrical potential that depends only on the concentration ratio.All the complicating or incalculable factors will cancel as they are the same for both sides.The direction of electrons flow will be that which tends to make more equal the two concentrations (from the more dilute to the more concentrated solution).