It’s more like parties switched bases; see Nixon’s Southern Strategy:

As the civil rights movement and dismantling of Jim Crow laws in the 1950s and 1960s visibly deepened existing racial tensions in much of the Southern United States, Republican politicians such as presidential candidate Richard Nixon and Senator Barry Goldwater developed strategies that successfully contributed to the political realignment of many white, conservative voters in the South who had traditionally supported the Democratic Party. It also helped to push the Republican Party much more to the right relative to the 1950s. By winning all of the south a presidential candidate could obtain the presidency with minimal support elsewhere.