Americans disliked the Soviet Union because the Soviets were communist. They wanted to be better than the Soviet Union to prove that our system (democracy and capitalism) was better than the Soviet system.
Communism is a system that is completely opposed to our system of government. In communism, the government owns all of the factories and other “means of production.” People do not have much freedom of choice at all. The government decides what kinds of things factories will produce, so customers don’t have many choices. People are generally not allowed to speak their minds the way we are in a democratic system. Americans disliked the Soviets because they disliked the ideas of communism.
During the Cold War, the US and the USSR were competing to see who could get more countries on their side. They each wanted to dominate the world. We felt that other countries would like our side if we were better than the Soviets. So, for example, we thought that our system would look good if our athletes could beat their athletes or if we could reach the moon before they did.
This is why Americans disliked the Soviets and wanted to be better than them.