Well, when you start breathing too fast (hyperventilating) it produces too much oxygen in your blood which makes you dizzy. When you breathe into the paper bag, that collects carbon dioxide which you have breathed out too much of when hyperventilating. So breathing in and out of a paper bag helps to bring the levels of oxygen and carbon dioxide in your blood back to normal again.

I know that's not a very good explanation but I hope it makes sense.