Why do bug bites itch?

Because if you are talking about a mosquito when it bites you, it puts its saliva in your blood stream to help the blood to keep flowing while it sucks its "Meal"....The saliva from the mosquito will itch where it bit for a while until the bite is healed.

Concerning other bugs, the infection from the bite will itch until properly treated and also it will itch during the process of healing.

I hope this helped! :)