Lets think straight and level.
Lift is equal both sides.
Now turn control a little to the left, lift (in the area of the aileron), on right wing increases, and on left wing decreases.
Turn a bit more and lift on right increases more and decreases more on left.
Keep turning the control and eventually the lift on the left wing may become zero and then negative, but will never be as negative as the right is positive, so the induced drag will be more on the right and hence the adverse yaw.
This is a simple explanation but should help.