What causes electrons to move from zinc to copper?
There is no reaction since the electrons will only flow one way in the wire. When you connect the zinc metal to the copper metal with a wire, there is a voltage potential generated between them, just like in a thermocouple.

From Wikipedia:


  Any junction of dissimilar metals will produce an electric potential related to temperature.  


In this case, we don't care about the temperature effects.  

When a salt bridge is placed between the solution, then the electrons are carried by the anions in the solution from one electrode to the other electrode.  

So the zinc metal releases electrons when it dissolves, which travel through the wire easily, to the copper. The $\ce {Cu^{2+}}$ in solution grabs the electrons from the copper electrode, adding copper metal to the electrode. Since there is now more negative ions in this solution, the anions move across the bridge to the zinc electrode. The anions attack the zinc metal, causing the zinc to loose electrons when it dissolves.  

I'm not sure what starts the reaction, nor why the zinc dissolves easily.