Some racists (e.g. Strom Thurmond) switched parties, but in general the Republicans were never as racist as the Democrats historically were. They didn't have to be. Popular support for racism has been on the decline for more than a hundred years, and the remaining racists no longer trusted the Dems so the Republicans never had to give them anything or even really promise to give them anything to get their votes.
This is kinda sorta true, but it neglects to mention that the "Southern Strategy"[1] was a real thing:
In American politics, the southern strategy was a Republican Party electoral strategy to increase political support among white voters in the South by appealing to racism against African Americans.... 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 that had traditionally supported the Democratic Party to the Republican Party.