It basically tells you how many connections (plane changes) you are allowed to make, and where.

In this case, AA only flies to Miami, Washington (DCA) and Dallas/Fort Worth from Bogota (BOG). Suppose you're in Boston and you want to get to Cartagena (CTG). You can change planes up to three times en route from Boston to one of AA's three U.S. gateways to Colombia. You may then change again at BOG for a flight to CTG. And you can change the same number of times when you return.