The piece is called Counting Sheep. When the handle on the box is turned, the sheep jump, and a counter is incremented.
A classic ratchet-and-pawl mechanism to drives the counter. Not an easy mechanism to engineer in paper!
Flying Pig offers printed kits and downloads. Order the kits you like, download the file (PDF), print them on card stock, cut them out, and glue them together with white glue. Each kit comes with complete instructions.
Check out Counting Sheep and the other paper animation kits.