This may be an overstatement, but in a sense the United States is the cultural world power. Why is this? Many people in foreign countries know American songs, American food, American movie stars, American movies, and so on. How did this come to be?
I can speak from my French point of view.
In France, American culture has been influent since the 30's-40's. More precisely, Californian culture. At that time, the state started being home to a great production of art, in literature and in cinema. It became popular in France with the detective books from Chandler and Hammett, and of course the Hollywood movies.
Since that time, French cinema got a very harsh competition from the American one. French movies are not as respected as American ones, they are typically comedies (the most successful French movies ever are comedies), day-to-day life representations or intellectual works. It also true for French books.
Why? a too easy answer is: because money. American books and movies have a much bigger market than the European ones and are produced by much more talents coming from all the country.
But it is not enough. I think the real answer is: because California. This state didn't just create a thriving economy. It created new ways of thinking, new dreams. It wanted to imagine things in its own way. With that spirit, it created a whole new culture.
It's something that happens only a few time in history. To give another example, it happened in Holland. When the inhabitants won their independance from Spain, they became so passionate, so optimistic, so proud, that they express their fervor in paintings and this resulted in one of the greatest painting movements in European history.