For an aircraft to stay in level flight, its lift needs to be equal to its weight. As weight increases, lift must also increase for the aircraft to remain in straight and level flight. We have learnt that by changing angle of attack or speed, the pilot can control lift.
If two aircraft with different weights are flying at the same speed, they must each be at a different angle of attack to maintain level flight. The heavier aircraft requires more lift and so will be at a higher angle of attack than the lighter aircraft.
Since the heavier aircraft is generating more lift, it must also be generating more induced drag. To maintain the same speed, the heavier aircraft will need a higher power setting to overcome this extra drag.
Imagine two aircraft with the same weight are flying straight and level at different speeds. The lift must be the same for both aircraft since their weight is the same.
The slower aircraft will need to be flying at a higher angle of attack to create the same lift as the faster aircraft.