Think of it as a 'purification conveyor belt' where dirty copper gets dumped into a pool at the anode and is reassembled perfectly clean onto the cathode mold.

Remember 'AC' for Anode-Crude (impure) and 'CP' for Cathode-Pure.

When an electric current is passed through the electrolyte, copper atoms from the impure copper anode lose electrons to form $Cu^{2+}$ ions and dissolve into the solution. These $Cu^{2+}$ ions migrate to the pure copper cathode, where they gain two electrons to become neutral copper atoms, effectively plating onto the pure copper strip. The overall net reaction is simply the transfer of pure copper from the anode to the cathode: $Cu_{\text{anode}} \rightarrow Cu_{\text{cathode}}$.

Always remember that in all electrolytic cells, oxidation (loss of electrons) occurs at the anode and reduction (gain of electrons) occurs at the cathode, regardless of the metal being refined.