The British Empire was at one time the greatest and most expansive Empire in the world. It covered such a large expansion of the globe that the common expression was "the sun never sets on the British flag." In typical Western European fashion, the British imposed their culture, and their language over their entire Empire. English became the official language of India, Pakistan, and Australia. It previously was the official language of the United States and Canada by reason of British settlement and governance. The pervasiveness of the Empire led to the pervasiveness of the English language. I have seen no evidence of the influence of the United States during or after World War II which led to this pervasiveness. English is widely spoken in Japan, in fact all street signs, train announcements, etc. are made in English and Japanese. However, the Japanese made a conscientious decision to emulate American culture long before World War II.
I agree with brettd. Two of the biggest nations in the world spoke English. Add to that the powerful role the United States had in the war and the vast amount of region that Britain controlled, and it isn't surprising that English became the predominant language. No other countries had that much power.
The British Empire also had something to do with it. Consider how many African and Asian countries were under British rule up until 1945, or in the case of Africa, until 1961 even. So the British have been spreading the language for centuries. All Americans did was accelerate it, and give it a less imperialistic motive: business.
Once you move beyond the first 20 years after WWII, you should also consider technological advancements in the United States. The United States leads the way with computer technology, email, the Internet etc. and these have created our global community. English is the most common 2nd language in the world because it is the way the world can interact with eachother using current communication systems.
I believe that this happened mainly after the end of WWII. In the time after the War, English spread because the US was so powerful.
During the '50s and '60s, the US was by far the richest and most powerful country in the world. Most countries wanted to be able to trade with the US, for example, and so English became the lingua franca for trade. This helped expand the reach of English.