Induction motors r much like transformers. Infact their electrical models r almost identical, the turns ratio for transformer being replaced by slip only.
The stator voltage creates a 'rotating' magnetic field which is the result of the pjhase difference & the physical orientation of poles. This field, when cuts the rotor (which consists of copper bars shorted by end rings) creates circulating currents within the rotor bars, resulting in a secondary magnetic field. This field sort of follows the stator & the rotor moves.
IN single phase motors, the field doesnt rotate but oscillates perpendicularly. This kind of field cannot provide the starting torque to the rotor & the motor becomes useless unless someone gives it a starting turn. To create a rotating field another set of windings is added, with the CAPACITOR in series. The capacitor results in a voltage phase shift in the second winding, the physical orientation of the stator & the starter windings creates a rotating field enough to provide the starting torque. Once the motor starts running, a centrifugal switch disconnects the starter winding & capacitor, since inertia is enough to keep the motor running smoothly in the oscillating field.
The capacitor can be selected to improve the power factor of the motor, & can be left connected as well, only this reduces the starting torque.