I am no aerodynamicist, and my experience is on hang gliders rather than microlights, but this is my understanding of what happens on hang gliders and I believe microlights are analogous.
A hang glider pilot is in a harness rather like a climbers harness, you can sit in it and the weight would be supported on your upper thighs and buttocks. But the pilot would normally, after take off, lean forward and some of the weight would then be supported by shoulder straps - this is for drag reduction and comfort, not aerodynamically mandatory, and not done with the very early harness models.
There is a strap connected from your harness via a carabiner to a loop which is permanently connected to the glider keel. Your weight is taken on this strap (breaking strain usually two ton, and there are usually two of them)
. This strap is usually near vertical, unless or until you start to pull a lot of G - not recommended. You are holding the bottom horizontal bar of an A frame rigidly connected to the glider. If you push the A frame to your right, you have effectively shifted your weight to the left. The strap is now dangling from the keel to a point a bit to the left of the centre of the horizontal bar, so no longer quite vertical, so your weight, about three feet below the centre of lift, is now a bit to the left of the centre of lift. If the glider is stiff you will have to hold this by muscle force until the wing starts to bank under the this offset load. Eventually you will turn left. If the glider is spirally neutral you will end up in a turn where the strap supporting your weight is once more perpendicular to the wing, but the wing is not parallel with the horizon, and the strap therefore, is no longer perpendicular to the ground. While many gliders are spirally neutral, some are spirally stable, some are spirally unstable, and some vary according to how you change the tension in flight. A modest degree of spiral instability (a tendency for turns to steepen), is desired for staying centred in small thermals - by hot ship pilots, not me. And spiral stability is definitely not a topic I can explain.
You ask why the increased drag on the left wing when you do this. The wing is effectively a sail, albeit somewhat stiffened by battens, the more you load it the more it will deform, and the increased deformation makes it less efficient and increases drag.
With a sailplane with a rigid wing, the greater the load, the more efficient the wing and the better the angle of glide - even though the actual descent rate increases. This is why in races they often carry water ballast.
If you put a heavier pilot on a hang glider,it will fly (and stall) faster, but at a worse angle of glide, as the wing deforms more and becomes less efficient. So if you shift your weight as above you are loading the left wing more, deforming it more, and increasing the drag more, on that side only.
Probably simpler to draw it but I am not sure how to attach a drawing.