The stinger usually gets stuck when they sting a mammal and the bee dies. When they sting other insects like other bees, the stinger may or may not get stuck depending on where they sting and how deep the stinger penetrates. I have watched bees sting each other in my observation hive and sometimes they can repeatedly sting other times they get stuck the first time.
It's the elasticity of the skin that causes it to catch and pull out. When they sting other insects it usually doesn't get caught but as ffrtsaxk said, sometimes they do. But mammals all have that elasticity so it usually gets caught.
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