You would probably expect a jumping spider to have well-muscled legs, like a grasshopper. But this isn't the case at all. Each leg on a spider has seven segments: coax, trochanter, femur, patella, tibia, metatarsus, and tarsus. Just as we do, spiders have flexor and extensor muscles, which control their movement at the joints between two leg segments.
Spiders, however, don't have extensor muscles at two of their six leg joints. Both the femur-patella joint and the tibia-metarsus joint are missing extensor muscles, meaning a spider cannot extend those parts of its legs using muscles. Jumping requires a full extension of the legs, so there must be something else at work when a jumping spider leaps into the air.
When a jumping spider wants to jump, it uses a sudden change in hemolymph (blood) pressure to propel itself upward. By contracting muscles that join the upper and lower plates of the cephalothorax, the jumping spider can effectively decrease the volume of blood in this region of the body. This causes an instant increase in blood flow to the legs, which forces them to extend rapidly. The sudden snap of all eight legs to full extension launches the jumping spider into the air!
Jumping spiders aren't entirely reckless, by the way. Before pumping up those legs and flying, they secure a silk dragline to the substrate beneath them. As the spider jumps, the dragline trails behind it, functioning as a safety net of sorts. Should the spider find it has missed its prey or landed in a precarious place, it can quickly climb up the safety line and escape.
Source: The Encyclopedia of Entomology, by John L. Capinera