What causes a voltage to appear across the secondary winding of a transformer when an AC voltage source is connected across its primary winding?
Why A transformer works because the changing current in the primary creates a changing magnetic flux in the core, and that flux links the secondary turns. By Faraday's law, a changing flux through a coil induces a voltage in it, and the shared-flux coupling between two coils is called mutual inductance. This is also why transformers only work on AC: a steady DC current makes no changing flux and no secondary voltage.
Watch out Capacitive coupling relies on an electric field between conductors separated by a dielectric, which is stray and unwanted in a transformer, not the mechanism that transfers power from primary to secondary.
Transformers are coils, and coils mean inductance: two coils sharing flux equals MUTUAL inductance.