The brake cable can be bent tightly around the anchor bolt, if the cable end was routed along the brake arm. That "crook" in the cable, plus any tab or washer, plus not loosening the anchor bolt enough, could cause the cable to stick in place. You need to free the cable end, loosen the bolt more, wiggle, yank hard, etc.

But why are you trying to release the cable from the anchor bolt?

You said the brake (pads?) rub the rim. That's usually because the rim is not true (not straight, wobbles from left to right as you spin the wheel). Or because the brake springs are misadjusted so one pad is always contacting the rim while the other pad is far away from the rim. If the brake pads are simply too close to the rim, you can usually adjust that with the barrel adjuster on the brake lever. Undoing the anchor bolt is one of the last things I would be doing.