There are a.c. in -ve and ve of same amplitude, so fan is not able to move in any direction, due to the inertia of motor. Now fan capacitor increase the amplitude in ve direction of a.c. and this makes the force on the motor in one cycle and in one direction. so fan is able to move.

The above answer is not very easy to understand and does not seem to be correct either.

What follows is a very simplified explanation because the full explanation requires an understanding both of electrical technology and some fairly advanced math. It is not possible to simplify the full explanation enough to be able to write it all down in a few paragraphs here. Whole chapters have been written in books to cover this subject!

The most common motor used in a normal household electric fan is a single-phase induction motor. When first switched on such a motor needs to use a “starter” winding, in addition to its main winding, so that a rotating magnetic field is created to make the motor’s rotor start to turn.

However a starter winding cannot do the job on its own, it needs a “Start-And-Run Capacitor” in series with the starter winding to create a rotating magnetic field. Without one of CBB61 capacitors the motor would never start to turn.

If you really want to understand how electric motors work the best thing to do is to go to a library and read some Electrical Engineering books about Electrical Machines.