That's an overstatement, which has been popularized by Democrats to distance themselves from their longstanding coalition with southern segregationists. The key Civil Rights Acts were passed from 1957 to 1968. The political alignments on various issues haven't changed much since FDR. Democrats were on the liberal alignment with respect to government regulation, business freedom, taxes, education, immigration, social welfare, religion, gun control, etc.
Contrary to your statement, the Civil Rights Acts were not a "precipitating event for politicians switching parties." That doesn't even make sense--why would politicians who were against civil rights join the party that much more strongly supported every Civil Rights Act from 1957 to 1968?
The realignment of southern democrats actually occurred much later. Nixon did not win a majority in any southern state--to the extent he won with a plurality, it was only because the Democratic vote was split between Humphrey and Wallace. In 1976, Carter won with the same east-coast south/north coalition that long voted Democrat; with Ford winning the west coast and mid-west. Reagan won almost every state, but his margins in New York were larger than his margins in Alabama or the Carolinas. Reagan did blow out Mondale in the south in 1984, but I'm not sure how much that tells us. Even by the time of Clinton, he won Louisiana, Tennessee, Kentucky, and Georgia, not to mention Arkansas.
I think the more accurate take is that the political realignment of the parties on "civil rights" issues happened more in the mid-late 1980s through 1990s. And it happened because the nature of the "civil rights" debate morphed over that time. The battle fronts during the 1980s and 1990s was not eliminating de jure and overt discrimination (the aim of the 1950s and 1960s legislation republicans supported), but measures like affirmative action, which sought to use the power of government to shape private conduct to eliminate existing inequities. That of course maps very cleanly onto longstanding republican versus democrat positions.
(I'll give another example of situations where political alignments change because the issue has changed rather than the "mix of platforms" of the parties. On the abortion front, for example, a significant amount of the debate has moved from talking about whether it should be legal at all, to talking about whether religious organizations should be required to provide healthcare coverage for them, whether the government should support them with public funding, etc. If you're a consistent libertarian, you might have found yourself more aligned with Democrats back in the early 1990s, but more aligned with Republicans today.)