Why does torque increase with rotor current in an AC induction motor?
The increase in torque can be explained by the following physical relationships:


The Torque that develops in a motor comes from the interaction between the stater's and rotor's magnetic fields.
The strength of each magnetic field is proportional to the current flowing through the stater and rotor.
The magnetic field of the rotor will attempt to align with the magnetic field of the stater(South to North and North to South), this is where the turning force exerted between the rotor and stater comes from.

Therefore if you push more current through the rotor a stronger magnetic field is developed and a greater turning force is exerted from the interaction of those magnetic fields.  This concept is more simply illustrated by looking at a brushed DC motor. Picture below from: https://www.globalspec.com/reference/10791/179909/chapter-3-ac-and-dc-motors-ac-motors-ac-induction-motor




Hope this Helps!