Joseph McCarthy became popular because many Americans (especially conservatives) were very suspicious of communists and thought that communists were likely to try to take over the US from within. Because McCarthy seemed to be a regular guy who was fighting elites who were probably communists, he was quite popular. This was especially true at first when the Korean War and the "fall" of China were still fresh in people's minds.
McCarthy fell largely because he overreached. He started to claim that people within the army were communists. When he did this, he ran up against a part of the government that most people respected. So now, instead of attacking egg headed elites, he was attacking respected military people.