If you've ever made—or seen someone else make—a loaf of bread from scratch, you've witnessed how something can grow exponentially. Over time, yeast goes through a reaction causing dough to rise and expand.
Compound interest works in a similar way. When you save money, you earn interest; when you invest it, you might make gains or earn dividends. In both cases, if you leave the interest, dividends, and gains in your account, they will be added to your principal—which is the amount you originally had in the account. If you earn interest or additional gains again, they'll be added to that combined balance.
This process can happen over and over again ... causing your original dough to grow over time.