Paper cuts hurt so badly, because paper does more microscopic damage to the skin, and there often isn't blood to protect nerves afterwards.
Most paper cuts occur on the fingers and hands, because it is the obvious part of the body that handles paper. The thing about the hands is that they are full of nerve fibers called nociceptors and very sensitive with sensory instruments. The nociceptors sense temperature, pain, and pressure.
There are more of these nociceptors on the hands and fingers than on other parts of your body and therefore hurting or damaging the hands tends to be a more intense feeling. Paper cuts are horrible to get, but only the hands. The same paper cut on a leg would barely be noticed.
The other thing that intensifies a paper cut is the fact that you can’t just quit using your hand until it heals. So, the wound continues to be bumped, rubbed, and altogether disrupted again and again.
The other thing is that paper is flexible, so when it cuts, it normally cuts at a wave instead of a straight cut like a knife would do. So, it does more microscopic damage to the skin.