An induction machine does not need an externally supplied rotor current to create torque. The rotating stator field itself induces the rotor currents.

Relative motion produces induction

If the rotating magnetic field moves relative to the rotor conductors, the rotor experiences changing magnetic flux. Faraday's law therefore induces rotor EMF and current.

Those currents interact with the magnetic field and produce electromagnetic torque.

Lenz's law and torque direction

The induced effect opposes the relative motion that created it. In motor operation, the electromagnetic torque therefore acts in the direction of the rotating stator field and tends to reduce the speed difference between the field and the rotor.

Why synchronous speed is special

If the rotor were to reach exactly the same speed as the rotating field,

$$n=n_s,$$

there would be no relative motion in the ideal steady state. The rotor would then see no changing flux, so no rotor EMF or current would be induced and no induction torque would remain.

A loaded induction motor must therefore rotate slightly below synchronous speed.

The amount of relative motion is quantified by slip.