The U.S. first became involved in the Cold War right before the fall of Germany in 1945. Even before VE Day had been declared, the Allied powers (Britain, USSR and US) had been trying to decide how to redraw the map of post-war Europe. The U.S. and the British had wanted to guarantee free and fair elections following the war, but the USSR had different aims. With a mind towards securing their eastern border, the USSR sought to dominate the politics of its neighbors, creating a buffer zone of friendly countries between them and the rest of their western neighbors. Following the end of the war, the USSR annexed several countries as Socialist Republics, creating what became known as the Eastern Block. The rest of Western Europe looked on helplessly as country after country fell under the sway of the Soviets.
The U.S. really became invested in the cold war during the Greek Civil War of 1947, when pro-communist forces sough to topple the monarchical government that was supported by the West. President Truman decided that it was in the best interest of the U.S. to stop the spread of communism and that the U.S. would spend money to achieve this goal. This policy became known as the Truman Doctrine, and set the U.S. on a path that would eventually take it to Korea, Cuba, Vietnam and many other nations.
The United States was the world power after the World War and and they were the only ones to have an atomic bomb at that time.