Why do you get a rash from a bee sting?

Firstly, let me stress that not all people are allergic to bee stings. For most people the sting is just a painful nuisance.
Bee venom is a complex substance. Its main active components are peptides (small proteins or parts of proteins) and these can trigger the immune system: the same mechanism that helps protect us from disease. In some people the immune system over-reacts, causing the allergy symptoms.