A centrifugal switch is used in conjunction with split phase and other types of single phase induction motors. Centrifugal cutout switches will open or disconnect the starting winding when the rotor has reached a pre-determined speed, and reconnect it when the motor speed falls below it. Without such a device, the starting winding would be susceptible to rapid overheating and subsequent burnout.
The capacitor in this case is a start cap, used to boost the motor up to run speed, and is not in the circuit once the centrifugal switch comes up to speed and clicks over to run. What causes these caps to blow is excessive strain on the motor, either by overloading or from other components that are binding. The caps get quite warm to the touch.