The most important thing to note with regard to this statement is the fact that Richard Nixon won the 1968 election quite easily in the Electoral College. He did not win a majority of the popular vote, but the bulk of the vote that he and the Democratic candidate did not win went to George Wallace, who was a conservative. This meant that the liberal candidate, Hubert Humphrey, won less than 43% of the popular vote while the conservative candidates won almost 57% of the vote. From 1968 to 1992, Republicans won all but one of the presidential elections. This is not the sort of evidence that would show that conservatism had died.
Conservatism remained quite strong in the United States, but it was muted for much of the decade. Nixon was right when he said that there was a "silent majority" of Americans who were tired of the racial strife and the counterculture and other liberal aspects of the 1960s.