Why does a smaller sprocket on a bike help you?

It's no guarantee that it will, it all depends on what you want to achieve. If you currently feel that you're strong enough to keep the pedals spinning like crazy, then going to a smaller sprocket means that the rear wheel will move more than before for each turn of the cranks - making you faster. But if you currently feel that pushing the pedals down is hard, then getting a smaller sprocket will only make that worse. If you want to be technical it's all about gain ratiOS.