The US earned this role in two ways.  First, by being the one major country in the world that was relatively untouched by World War II (in terms of its industrial capacity).  Second, by being willing and eager to take on the Soviet Union in the Cold War.

After WWII, most of the other major nations had been badly bombed, severely damaged by invasion, or both.  This made these other major countries much less capable of affording a large military.  It also mean those countries needed economic help, help that only the US was in the position to provide.

In addition, the US became a world superpower because it did not just sit back safely behind its borders.  Instead, it involved itself in trying to prevent the spread of communism.  This got the US involved in affairs in every corner of the globe.

America's industrial capacity and its Cold War willingness to get involved in the affairs of countries all over the world, made the US a superpower.

How did the US become a superpower in the 20th century?

This is a great question. A little history would help here. 

First, the 20th century was destructive for many powerful nations on account of wars. For example, Europe was ravaged by warfare as well as the Soviet Union. There was much infrastructural damage and the need to rebuild after World War I and II. The United States did not have to deal with this, because the war was fought in Europe and not in North America. This fact alone put the United States ahead. 

Second, the rest of the world - Asia and Africa were still weak. So, they could not really challenge America. Hence, we can say by default America became a super power. 

Third, as the economy grew after the war, America continued to climb in power, wealth, and prestige. In light of this, soon America became a super power.