Fan speed switch: why OFF :: HIGH :: MED :: LOW?
Fans of that type have induction motors with two windings with a capacitor in series with one of the windings. For every individual motor design there is a certain capacitor value that allows the motor to develop maximum torque and operate at the maximum corresponding speed. Smaller capacitor values are used to reduce the torque so that the load overcomes the motor's torque and slows the motor down.

When the motor is turned on, the motor must overcome the static friction of the bearings and get the fan moving. It must then accelerate the motor inertia. That means that using the highest torque setting is desirable for starting the motor. Once the fan is moving, the torque can be reduced for lower speed operation.