There are a couple things at play here, some involving the paper, some involving your skin.
For one thing, what part of your body comes in contact with paper the most? Right, the majority of paper cuts happen on the fingers and hands. Your hands are pretty complex sensory instruments, and they’re absolutely jam-packed with nerve fibers called nociceptors. These guys sense temperature, pressure and pain, and there are more of them per square inch in your hands and fingers than most other parts of your body. Injuries there are noticed much more than similar injuries elsewhere. The same small paper cut on a less nerve-dense area, such as, say, your leg, won’t send nearly as many pain signals to your brain.
To make matters worse, it’s not like you can just not use your hands for a few days after every paper cut. You’ve got stuff to touch and things to pick up, so while the cut heals, the skin keeps moving and the edges of the wound tend to get pulled apart, delaying healing and prolonging the pain.
All right, sure, your hands are very sensitive, but then why do paper cuts hurt more than some cuts made by other objects, like knives? Well, the blade of even a fairly dull knife tends to be more straight and sharp than the dull and flexible edge of a piece of paper. When a knife cuts your skin, it leaves a relatively clean cut compared to paper, which will flex a little and do more microscopic damage to the skin. Paper also makes a more shallow wound than most other cutting injuries. A shallow cut on the surface might only bleed a little, or not at all. Without a blood clot to protect them, the nerves around the cut are exposed to air and other irritants, which can make the pain more noticeable and longer lasting.