Why a pure single phase induction motor does not have starting torque but has running torque?

A single-phase induction motor can be considered as running on two oppositely-rotating magnetic fields. In normal running just below synchronous speed, the slip is perhaps 0.05 on one of these rotating fields and 1.95 on the other. The first field supplies the major part of the torque.

When the motor is stationary with power supplied, both fields have a slip of 1 and they produce contrary torques so the motor remains stalled.

If the motor is spun in one direction, the slip becomes unbalanced and there is a net torque from the direction with the least slip, so the motor runs up in that direction.