Exactly what the switch looks like varies, but after taking the wired end the motor off, it will be found mounted on the axle something like this:
As the motor gets up to speed, a weight, or pair of weights, pulls against a spring and eventually move far enough to open a contact. Every time the motor starts, the contact sparks and they eventually wear out. Or the lever gets out of adjustment so the contact doesn't move far enough to open and close correctly. If the contact can't close when the motor stops, the run winding gets no power, and the motor won't start.
The contact is usually easy to see, and it may only need a clean or adjustment. If it's burned out might be possible to replace the contact points, or to source a replacement centrifugal switch. It depends on the age, make, and type of motor. I'd expect Bridgeport spares to be available, but not necessarily cheap!
The centrifugal switch and capacitors are a weak point in single-phase motors and they vibrate. The advantage of single-phase motors is they can be plugged into ab ordinary mains socket: everything else about them is shades of grey! If the motor has to be replaced, consider 3-phase and a VFD.
Dave