To put it simply, aluminium oxide undergoes a redox reaction in the electrolyte.
At the cathode: Aluminium is reduced and does not mix with the electrolyte but falls to the bottom of the cell.
Al³⁺ (l) + 3e⁻ → Al (l)
At the anode: Oxide ions are oxidised and oxygen is given off.
2O²⁻(l) - 4e⁻ → O₂ (g)
Hopefully this answers your question.