Very straightforward.

Let's start with a wing having no camber, flying along minding it's own business at 2deg angle of attack, flying level and supporting the weight of the airplane.

The pilot decides to do a roll to the right. The right hand side aileron deflects up, changing the camber of the right side wing and effectively reducing its angle of attack to (say) zero. The left hand side aileron deflects down and the effective AA of the left side wing increases to (say) 4deg.

Induced drag is nature's "price to pay" for generating lift. There are no "free lunches" in nature, and the result of generating more lift is more induced drag. The right hand wing, which has its AA reduced, generates less lift than before and so its induced drag is reduced. The left hand wing, which has its AA increased, generates more induced drag.

We see from this that even if there was no nett change in total drag on the wing, there is a redistribution of drag towards the rising wing (left hand side). This will cause a yaw of the aircraft in the direction opposite to the intended turn.

This simple explanation ignores the effects of spanwise flows, vortices, fuselage interactions, prop wash, etc. etc. Purists stay away!!

How to fix it (in plain english)?

We see that the problem is the up-going aileron which reduced the lift on that half wing and consequently the drag, causing a drag asymmetry. If we made this aileron go up by a large deflection then two things would happen. Firstly, because the aileron is an inefficient camber changing device, it will add extra drag beyond that of the other aileron. Second, if the deflection is large enough, we may be able to get the wing to develop substantial negative lift (ie. push down) and consequently generate lots of drag. Both of these effects will tend to neutralize the drag asymmetry.

So, the method is to arrange for the up-going aileron to go up further than the down-going aileron goes down (jeeeez, there must have been a better way to say that!!).

There is a problem. It doesn't work when the airplane is inverted. Well it does work, but it works against us. See, the problem exists because of gravity. In level flight the wing starts at some finite positive AA, because it needs to support the weight of the airplane. If it were not for this then there would be no asymmetry and differential would not be required.

The best solution is to eliminate gravity. I'll let you know when I succeed!!