Why is the single phase motor not self starting ?

For an induction motor to be self starting, the stator needs to have a rotating magnetic field.  In a three phase induction motor, the rotating magnetic field is created by the fluxes of the three phase windings  which are displaced by 120 degrees.

This rotating field induces an emf in the rotor.  This results in the flow of current which results in the magnetic field in the rotor.  This results in the magnetic field of the rotor trying to catch up with the stator and the rotor starts to rotate.

In a single phase motor, the magnetic field in the stator is the result of only one phase.  Thus the magnetic field produced is pulsating.  The field is not rotating.

Hence, the rotor is pulled in opposite directions by the stator magnetic field.  This results in the rotor staying the staying position.  Hence, the single phase motor needs an auxilliary winding which produces two magnetic fields displaced in time which provides a magnetic field.