Wasps do not die when they sting. Wasps have barbs on their stingers that can be retracted, allowing them to safely remove the stinger without injuring themselves.
A wasp's stinger is actually an egg-laying apparatus called an ovipositor. It doubles as a defense mechanism and is capable of injecting venom from venom sacs through the stinger if needed. The ovipositor is a female-only trait, making males incapable of stinging.
Wasps should not be confused with honeybees, whose stingers are removed along with other organs when used. Honeybees can only sting once because of this, as they die once the stinger is deployed.