The United States and the Soviet were the two dominant superpowers left in the world and they had competing visions for what the world should look like.
The United States was a liberal democracy with a capitalist economic system while the Soviet Union was a communist country. Both were trying to export their vision around the globe.
So, during the Cold War, both were jockeying to be the MOST powerful country in the globe and did so by influencing countries around the globe.