Successor means "one who takes the place of", or "heir". In the context of succession, "succeed" means "take the place of", or "inherit". For example, Russia is the successor state to the Soviet Union. (Russia inherited the Soviet Union's U.N. seat, the nuclear arsenal, et cetera.)
In the original poster's example, Britain was the world's greatest naval power from the War of the Spanish Succession through World War I. The United States became the world's dominant naval power during World War II. In other words, the United States inherited the role of naval hegemon.
In the context of naval power and air power, "force projection" is the ability to make war at a great distance from one's homeland. Not only does the United States "project" military power, it also has a great deal of economic and cultural influence. (Hollywood's movies, television shows, and music have been called "cultural weapons of mass destruction" by American conservatives.) Thus, America "puts forth" economic and cultural power to most parts of the world.