When we switch on the electric fan why does it first moves slowly then moves fast why does not it move fast from the beginning?

The fan motor provides a torque τ which has to accelerate α the fan blades whose moment of inertia is I:τ=IαGiven how long it takes for the fan blades to stop the frictional torques must be fairly low and so the torque applied by the motor to keep them going must also be low. With the relatively small torque rating, even if the motor applied maximum torque when starting from rest, it would still take a noticeable period of time for the blades to reach their final rotational speed.The capacitors are there to make sure the motor rotates in the right direction when switched on and/or to allow the speed of the motor and hence the blades to be controlled.

A simple answer is that, any body at rest has inertia of rest as per Newton`s first law. So to set the fan in motion form resting position, more % of torque is consumed to overcome inertia and static friction. Once inertia is overcome, torque will be completely used to produce circular motion