How does take off mass impact induced drag? In this answer to another question, it was stated that: 


  Take-off mass: If you take no payload, the aircraft will produce less induced drag and reach a slightly higher range


How does a change of weight inside the same shaped container increase induced drag, since the air doesn't come in contact with the cargo?
Induced drag is caused by lift. (historically, Induced Drag was named "lift induced drag").

If weight(mass) of the aircraft decreases, the lift required to fly the aircraft decreases. And the drag associated to that lift also decreases.

The quoted statement is true, but "slightly" can be discussed. Because the increase in range depends a lot on how much payload the aircraft could carry. (If it's %50 of the max takeoff weight, the range could increase a lot).