After WWII the Soviet Union tried to influence countries with communism ideas, also creating smaller countries on their side in some cases. Sense communism has failed a lot in the past America tried to keep some of these countries from communism. I think the U.S also intervened because if war did break out, the U.S wouldn't want as many allies on the Soviet Unions side.
There also was the space race sense new rockets were made after WWII people thought of going into orbit and even getting to the moon. Although i think the Soviet Union had the first man in space, the U.S got the first man on the moon.
I'm not an expert but i think because of the Soviet Unions economy is what led to the U.S prevailing in the end of the space race.
What led to the cold war was mainly trying to become the world super power after defeating Germany because the U.S and the Soviet Union disliked each other, but knew Germany was a bigger problem.