You have to remember how the squirrel cage is constructed. It is basically a tube of iron rings laminated or riveted together attached to the rotor shaft that is attracted to and is dragged around by the rotating magnetic field of the stator. A double cage rotor has a second tube of iron rings inside the outer tube. This allows the creation of a stronger magnetic attraction between the rotor and the rotating magnetic field of the stator without significantly increasing the weight or diameter of the rotor and still allow the passage of cooling air through the motor.
This increased magnetic field will increase the amount of both starting and load torque the motor will produce without changing the motor input voltage or rotor size.
Double cage rotors are commonly used in variable speed and reversible motors such as fan and winch motors where high torque and low starting current is required and where a wound rotor motor cannot be used.
Troubleshooting 101 "Don't over think it - K.I.S.S. it"