What is the purpose of a relay between a switch and motor?
The relay is likely connected to an auxiliary motor winding that has a different number of turns, a different wire size and resistance compared to the main winding. That would act to start the motor similarly to capacitor starting. There are several possibilities for the motor design details. It is not possible to completely and accurately explain the motor design without close examination of the construction. Photographs might help, but that might still not be enough to determine exactly how the motor works.

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Note that the relay has relatively few turns of heavy wire. I believe it is a current relay, wired in series with the main winding. When the motor is first turned on, a high current flows in the main winding through the relay coil causing the relay to connect the auxiliary winding. Since the aux winding has a different inductance and resistance, its current is phase shifted compared to the main current similarly to the phase shift that a capacitor would cause. That defines the direction of rotation and provides starting torque. When the motor approaches full speed, the current drops and the relay disconnects the aux winding.