Why do you get dizzy when you blow up a balloon?

You are "hyperventilating" -- exhaling repeatedly into the balloon is creating an imbalance in your lungs, as not enough carbon dioxide is being retained. The excess of oxygen can make your blood alkaline, and make you feel lightheaded or dizzy by constricting the flow of blood in the brain.

Breathing a few times into a paper bag, thereby keeping more carbon dioxide, will normally reverse the condition.