Fans use single-phase AC motors (induction motors). Unlike 3-phase motors, a single-phase supply alone cannot create a rotating magnetic field it just creates an alternating field, which can’t start rotation. It is connected in series with the starting winding of the motor. This creates a phase shift (≈90°) between current in the main winding and the auxiliary winding. The two currents are now out of phase → produces a rotating magnetic field → the motor starts spinning. Once spinning, the motor keeps running due to inertia and the rotating field. In short: Capacitor provides the torque needed to start the fan.