hi ,
as you can see in the drawing you find , you have 1 coil of motor supplied directly . The other coil is supplied in series with capacitor . The capacitor has greater value at start (2 in paralel ) and lower value after startup.
Why?
The first coil is for supplying ac current in the rotor by induction . Like in a transformer just that the primary is the coil from stator and the secondary is the rotating coil in rotor. The rotor coil is short circuited so it passes current thru it and thereby it has own magnetic field, same phase as the one who created it. In order to rotate this structure a second magnetic field is required so that the interaction between them can produce torque (remember when playing with magnets ?). This is second magnetic field is created by the other stator winding in series with the capacitor.
Why is this second coil supplied through capacitor and not directly like first coil ?
Because without phase shift the magnetic fields of both coils (actually 3 coils) would be same and no torque would be produced. The capacitor creates phase shift of current in the second coil so it's magnetic field is not the same. At startup bigger phase shift is required since the rotor is at standstill .Actually interaction between fields is maximum at 90 degree . (quadrature)
When only running capacitor is available , and when mechanically possible , the startup can be made by hand with a rope. First a rope, previousely rounded on the rotor is pulled so it starts rotating , than the power supply is applied while the motor is in rotation , than the mechanical load is coupled . Actually this simple startup was used to avoid startup current when no startup circuit was available . At smaller motors was enough to hit the pulley in a direction , after the motor accelerated , the load could be coupled.
Ps: if no centrifugal switch available , you can use a manual switch to start up the motor . Just don't forget to decouple the switch . If the startup capcitor remains on, the motor will overheat because of no slipping between fields , the second stronger field (the one which creates quadrature) will slow down rotor