To understand the set up better, read the thread on "Capacitor Selection".
The "Silly Old Duffer" post of 21:27 on 10/09/21023 shows very clearly the wiring circuit.
If you set the motor running, and then switch off, as it slows down you should hear a click.
This is the centrifugal switch remaking contact.
I am not an electrician, but my understanding is that when stationary, the centifugal switch makes contact, bringing the capacitor into circuit.
The start windings are then "live", but being fed via the capacitor, are out of phase with the main running windings.
This results in a torque which starts the rotor moving. As the motor speeds up, the centifugal switch opens taking the start windings out of circuit. The motor than runs on the "slip" between the magnetic field produced by the main eindings, and the field induced in the bars of the rotor.
In a motor that has seen use over a number of years, the capacitor could have failed, or the centifugal switch contacts may be burned and no longer making contact.
Replacing the capacitor (Note the comments about the voltage rating, it needs to be rated well above 250 volts ).
It is possible that no direct replacement centrifugal switch is available. If this is the case, either the existing one has to be repaired in some way (IF this is possible ) or a switch that is available, modified to fit. Again IF this is possible.
If you are unsure about what you are doing, involve someone who really does know what they are about.
You should be able to find a company that specialises in motor repairs, local to you.
When swarf got into the mmotor on my Myford, a local company managed to repair the damaged switch quite easily.
230 volts could prove fatal, or cause a fire.