The United States emerged as a world power, a "policeman" for the world, as it acquired interests, especially economic interests, around the world. One event that contributed to the development of this role was the Spanish-American War. This conflict expanded American influence in Latin America and the Caribbean by eliminating Spain's presence in the latter region and simultaneously expanded American interests in the Pacific by wresting the Philippines from Spain. Henceforth, even in periods of relative isolationism (like the 1920s) the United States had truly global interests.
The second event was World War II, or more accurately the aftermath of World War II. With Germany destroyed and Great Britain exhausted by war, the United States was, along with the Soviet Union, a dominant superpower. Through the newly-formed United Nations, the United States embraced a global role that included rebuilding Europe, a so-called "police action" on the Korean Peninsula, and the establishment of alliance systems and economic agreements that sought to promote peace and, depending on one's perspective, either maintain security or foster American hegemony over the Western world.
Finally, the aftermath of the Cold War, which ended with the beginning of the 1990s, left the United States as the world's only superpower. Though the rise of China and the return of Russia as a global force, along with the attacks on 9/11, have altered this role, the period since the end of the Cold War has seen repeated interventions, both military and diplomatic, around the world. These include military interventions in the Balkans, invasions and extended conflicts in Afghanistan and Iraq, and many other actions. While the wisdom of state-building and intervention in general remain a contentious subject in the United States, the role of the nation in world affairs is indisputably a crucial one.