"When paper cuts into the skin, chemical-coated fibers as well as bacteria and tiny particles remain in the wound and stimulate pain receptors in the skin.
"Because the cut is usually small and shallow, the skin on either side of the wound closes quickly, trapping the fibers and other particles inside. The result is a good deal of pain, and since the closed wound doesn't allow for much bleeding, the pain seems entirely out of proportion to a cut that you can barely detect."
(A Moment of Science: Paper Cuts)