Festinger's theory arose from his observations of a Wisconsin-based flying saucer cult of the 1950s whose prophecy of universal destruction failed to come true. The cult prophesied a vast flood would soon kill everyone on Earth except for the members of the cult, who would be carried away by flying saucers. Before the predicted flood, the cult was very secretive and very reluctant to speak to the media or make converts. After the predicted flood, they stopped being secretive and spoke very eagerly to the media.
Leon Festinger, who studied the cult, explained this precisely by the fact that their prediction failed: there was no global flood and no flying saucers arrived to carry the cult to salvation. The cult was ridiculed, and though they had an explanation for the failure of their prophecy -- on the night in question their prayers "had spread so much light that God saved the world from destruction" -- nobody took it seriously. This is why Festinger suggested that the cult became fervently evangelistic. The only way for them to reverse their humiliation was to convert other people to their beliefs. If everyone believed, no one would laugh. However, this did not work: after such a spectacular failure, the cult predictably failed to convert anyone. For further information on the cult and its behavior, see Festinger's When Prophecy Fails: A Social and Psychological Study and "When Prophecies Fail".