In the decade or so after WWII ended, Americans came to fear the spread of radicalism (more specifically, communism) in the US. This fear came about largely because of the spread of communism overseas.
Right after WWII, the Soviet Union took over much of Eastern Europe. The Soviets were explicitly dedicated to spreading their ideology and many Americans feared that the Soviets would do so by winning the hearts and minds of people in the US. Since the Soviet Union had been an ally in WWII, there were still many Americans at that time who thought well of the USSR. Other Americans feared that these communist sympathizers would help to promote communism in the US. This fear was manifested in the success of Sen. Joe McCarthy's claims that he knew of many communists in government and in the support for things like the HUAC hearings on Hollywood.
In the years that followed, it seemed that communism was on the move. The communists took over China and launched an invasion of South Korea. Communists were fighting the French for control of Vietnam. The Soviets got nuclear weapons. All of these things made it seem as if communism was succeeding in its goal to take over the world. Because of things like this, many Americans feared communism and a red scare arose in the US.