Catapults are tools that let us throw things farther than we would be able to just using raw muscle power (and jumping is just throwing yourself using your legs).
A bow and arrow uses exactly the same principle as a catapult (and is a bit easier to sketch), so here is an animated cartoon which illustrates how they work.
The key point is that the bow, like the elastic in a catapult, acts as an intermediary energy store. It absorbs energy at low power from the muscles, and delivers it at high power to the arrow.
Here is a video of the catapult mechanism in the knee joint of the back leg of a grasshopper, filmed while the animal makes a defensive kick (which uses the same mechanism as a jump).
Notice how the black bit in the joint bend before the leg extends - that's the equivalent of drawing back the bow in the cartoon above, or the elastic of a catapult. The technical name for the black bit is "semi-lunar process" (because it is approximately half-moon shaped!). They occur on both sides of the knee joint.
In the slow-motion video below (1000 fps), the grasshopper was persuaded to kick by tickling it with a small paintbrush.
Video courtesy of Prof. M. Burrows, University of Cambridge.