You have to be careful. In the system you show, the electrons are not supplied to the anode by the anions - they are spectators. The electrons come from oxidising the zinc metal to zinc cations:
Electrons flow through the circuit to the cathode, where they reduce copper cations:
Ions migrate through the salt bridge to maintain electrical neutrality. Thus you will find that the concentration of both anions and cations increases in the anode solution, and decreases in the cathode solution (in this example).
This is an example of a galvanic cell (as in a battery), where a spontaneous chemical reaction creates an emf and causes current to flow through an external circuit. The release of electrons at the anode gives it a negative potential, and the consumption of them at the cathode gives it a positive potential (relative to each other), so electrons flow from anode to cathode. In a galvanic cell the anode is the negative electrode and the cathode is the positive electrode.
In an electrolytic cell, the application of a current from an external power source causes a chemical reaction to occur, e.g. the electrolysis of water to hydrogen and oxygen. In this case electrons are pumped by the power source into the cathode, where they cause a reduction reaction, and pulled from the anode, where an oxidation reaction occurs. Therefore in an electrolytic cell, the anode is the positive electrode (attached to the positive terminal of the battery) and the cathode is the negative electrode.
The constant thing between galvanic and electrolytic cells is that oxidation takes place at the anode; reduction takes place at the cathode.