The fan has an AC single phase motor. The problem with single phase induction motor is that there is only  one phase on the stator winding. Hence the resulting magnetic field in a single phase  motor does not rotate. For starting  any type of motor there must be resulting magnetic  field  to create torque to rotate the rotor .In single phase motor without capacitor the field created by stator winding is positive in half cycle and negative in other  half so resulting net field is zero so torque is also zero. Now when capacitor is added to the additional auxiliary winding of stator it creates field which leads by 90 degree to main winding  field so we get two field which are 90 degrees to each other, resulting a  rotating field responsible for start of the fan motor. The capacitor used to start the motor after start  may be taken out of  circuit because as rotor is rotating each half field of stator creating field in  rotor in same direction so producing net field not equal to zero and motor can run continuously if we do not switch off the power supply to the motor.

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