In double squirrel cage motor outer cage rotor winding is of high resistance and low reactance. Inner cage winding is of high reactance and low resistance.
At the time of starting rotor frequency is equal to the stator frequency and there by the reactance of the inner cage winding is comparatively high (XL = 2πfL) because it is linking more inner winding than the outer winding. So the impedance of inner cage winding is very high. Hence the current flow through inner cage winding is very less comparing to the outer cage winding. That is a very high ratio of current is passing through the outer cage winding at the time of starting and there by produces very high starting torque.
When the rotor starts running the speed of the motor can be increased and the slip will be decreased and there by the rotor frequency (‘S’ = fr / f). So that in the running condition the reactance of the inner cage decreases to the lowest value and hence the Impedance (XL = 2πfL). So the current in inner cage winding will be comparatively more than the outer cage winding at the time of running. So now inner cage winding produces more torque than outer cage at the time of running and the motor running torque is good enough.