Why do we need to Split the Stator of Single Phase A-synchronous (Induction Motor) unlike the 3 Phase Induction Motor?
A 3 phase induction motor has 3 different currents flowing through it... these 3 cause 3 different magnetic fields that are oscillating at 120 degrees (ideally) apart, the interaction of these 3 magnetic fields in the air gap causes a rotating magnetic field, this magnetic field rotates at a speed we call synchronous speed, which is also the maximum speed the motor can reach (and will never reach due to losses). This rotating magnetic field pulls the rotor and starts the motor.
A single phase motor only has 1 current flowing in it, therefore there is no rotating magnetic field only and oscillating one therefore the motor is not self-starting. Usually single phase induction motors that are not split need to be helped to start even by just spinning the rotor a little with your hand to get it out of inertia.
Splitting the stator winding and positioning it in a specific manner allows for a slightly lagging/leading current to flow in that split winding. You know how 2 different magnetic fields that are oscillating in the motor... this allows for a rotating magnetic field that will pull the rotor and cause your single phase induction motor to be self-starting.