Think of it like a laundry machine: the impure anode is dirty clothes, the electrolyte solution is water/detergent, and the pure cathode is the final clean garment where the 'cleaned' copper accumulates.

Remember 'PANIC': Positive Anode, Negative Is Cathode.

During electrolysis, copper atoms from the impure anode lose electrons to form $Cu^{2+}$ ions which enter the electrolyte solution. Simultaneously, $Cu^{2+}$ ions from the solution gain electrons at the cathode to form pure metallic copper. The process is represented by the reactions: Anode: $Cu(s) \rightarrow Cu^{2+}(aq) + 2e^-$; Cathode: $Cu^{2+}(aq) + 2e^- \rightarrow Cu(s)$.

Always remember that in any electrolytic cell, the cathode is the site of reduction, and the anode is the site of oxidation. This rule applies universally across chemistry.