According to Everyday Health, bedbug bites itch because of an allergic reaction to the saliva left on the skin by the bedbug. The immune system releases histamines in response to the bite, and histamines cause the area directly around the bite to swell and become itchy.
The American Academy of Dermatology explains that bedbug bites do not itch immediately because it takes time for the body to become sensitive to the bite. The allergic reaction typically produces mild itchiness in most people, but can be severe in others. Mild itchiness can be relieved with an over-the-counter antihistamine. Serious reactions such as fever, difficulty breathing and irregular heartbeat require immediate medical attention.