They charge more in fees, and also pass on more as cashback/bonuses/prizes to customers.

For example, a visa/mastercard might be able to kick back 1~2% for cashback. American Express might be able to go 2~3%.

This is also why a lot of places don't accept Amex, the fees are higher and from the vendor's perspective they offer nothing in return.

PS - The whole credit card system is too expensive, and inefficient, but that's what happens when three or four companies run the whole game.