If you actually dig into the financial statements, they bring in revenue from interest from people who do pay, but that is roughly cancelled out by money lost from the smaller portion of people who don't pay and by the credit card company having to pay interest to borrow the money in the first place to be able to loan it out to their customers. There is very little actual profit from the interest side of the credit card business.
The actual profits are from the fees, mostly the transaction fees. Those transaction fees are paid no matter what if you use the card. The transaction fees are paid if you pay off your card in full. The transaction fees are paid if you pay interest. The transaction fees are paid if you never pay your debt. This is since the transaction fees are paid buy the store owner and not by the credit card user. The more they hook you into using the card, the more profit they make. Hence, the cash back, mileage points, or whatever just to hook you into buying more and giving them that sweet transaction fee profit.