Why does paper become uneven when soaked and dried?

Paper is made of many, many tiny fibers. When you wet a piece of paper, those fibers absorb water and swell up. As they dry, the relaxed fibers don't go back to exactly where they started. Some will have shifted or "untangled", moving out of the plane of the original paper. So yes, the paper actually does become slightly bigger because the fibers have stretched.