There are two major reasons for this, WWII and the Cold War.
WWII helped make America the leader of the West because it devastated the other countries that could have laid claim to that title. France, Germany, and Britain were all heavily damaged in the war and could not come close to the economic production of the United States for years after the war. The fact that the US was economically much more powerful than any other country after the war was instrumental in making it the leader of the West.
The Cold War contributed to this. The Cold War made other Western countries very afraid of the Soviet Union and of communist take over. This made them more likely than they otherwise would have been to follow the lead of the US. The US was the country most able to protect them from the Soviet Union in the years after WWII and therefore it came to be the leader.
The economic devastation caused by WWII, and the fear of communism after it, made the US the leader of the Western world.