Because it is made of plant cellulose, so it's tougher than it appears. It's made from wood, but pressed very thin. So thin that it can slide between the cells of the human skin so, if moved in a slicing motion, is strong enough to break through the cell walls of the skin and tissue.
A moving sheet of paper has a fair amount of momentum and strength in compression. When it hits your skin it will impart a force. This force acts on a very small area (the paper is sharp) and consequently applies a very large pressure to the surface of the skin. This is easily sufficient to penetrate the skin and cut it.
The same argument applies when you strike your skin against a stationary sheet of paper.
The best way if looking at it is take a drawing pin and press it between your fingers you'll notice the smaller end is sharper and more painfull because all the pressure is on such a small surface area same with paper