Oxidation can also be described using electrons. Oxidation is the loss of electrons and reduction is the gain of electrons. A helpful reminder is OIL RIG: oxidation is loss, reduction is gain. When a metal atom forms a positive ion it loses electrons, so the metal is oxidised.
Displacement reactions show both changes at once. When iron displaces copper from copper sulfate solution, the ionic equation is Fe + Cu2+ → Fe2+ + Cu. The sulfate ions do not change, so they are left out. Zinc behaves in the same way: Zn + Cu2+ → Zn2+ + Cu.
We can split the ionic equation into two half equations. Fe → Fe2+ + 2e- shows iron atoms losing electrons, so iron is oxidised. Cu2+ + 2e- → Cu shows copper ions gaining electrons, so copper ions are reduced. The electrons lost by one species are the electrons gained by the other, and the charges balance on both sides.