When you buy a “Honda Accord” car, you do not buy from the company Honda. You buy it from a dealer.
Credit card industry works similarly.
Visa and MasterCard own the network. Processing networks interact with Visa and MasterCard to process the credit card transactions. This business is all about volume of transactions. So it is hard for a new player to jump into market and to build a credit card processing network. Some of the companies with processing network are First Data, Chase, TSYS, Global payments etc.
These credit card processors sell access to their network to resellers. The resellers in turn sell the services to others. This is the reason why you find so many companies offering credit card processing services.