In earlier times, people were tortured until they confessed to crimes they may not even have committed. The guarantee against self-incrimination makes sure that unfair pressure cannot be used to make a person confess. In the 1950's, this section of the Fifth Amendment sparked a public uproar. Notorious criminals accused of having ties with the underworld claimed the Fifth Amendment protected them from having to testify. If they had denied those accusations, they probably would have been found guilty of 'perjury,' or lying under oath.
Perjury is punishable by law.
If they had told the truth, they would have risked punishment. Instead, they refused to testify on the grounds that they might incriminate themselves. Law enforcement officials had to come up with evidence against them or free them. The right to remain silent also protects innocent people. During the 1950's, the government became concerned about Communists.