I'm not sure that anyone is responsible for this shift; I rather suspect that politics is stochastic more often than planned. I'm not sure that it is possible to give an answer that a panel of objective observers would agree with. With those caveats in mind, I'd offer the following description of events.
I think the dominant player is FDR. FDR built a political machine that dominated US politics.
African Americans moved into the Democratic Party during Roosevelt's time. After Roosevelt took office in 1933, New Deal legislation sailed through Congress at lightning speed. In the 1934 midterm elections, 10 Republican senators went down to defeat, leaving them with only 25 against 71 Democrats. The House of Representatives was split in a similar ratio. Wikipedia's history of the Republican Party.
As the last couple of sentences point out the Democratic party grew at the expense of the Republican party. The few minutes of research I have available to me don't permit me to discover whether the GOP tried to preserve the party's record on civil rights, what strategies (if any) they tried, and why those strategies were ultimately ineffective. I suspect without any particular evidence that the issues that brought FDR to power and kept him there were, in the short term, perceived as more important to critical segments of the electorate than civil rights.
Nelson Rockefeller and Barry Goldwater are cited as the visionaries for the resurgence of the Republican Party. Both supported civil rights, and Rockefeller had a fairly strong and positive civil rights record. Goldwater in his 1964 election campaign made a strategic decision that federalism and states rights were more important than the Civil Rights Act.
In 1964, Goldwater ran a conservative campaign that emphasized "states' rights". Goldwater's 1964 campaign was a magnet for conservatives since he opposed interference by the federal government in state affairs. Although he had supported all previous federal civil rights legislation and had supported the original senate version of the bill, Goldwater made the decision to oppose the Civil Rights Act of 1964. His stance was based on his view that the act was an intrusion of the federal government into the affairs of states and that the Act interfered with the rights of private persons to do or not do business with whomever they chose. Wikipedia attributes this assertion to Donaldson.
You could therefore argue that Goldwater is responsible for the shift, but I think that oversimplifies the process of politics.
Note with emphasis. I've done my best to be as neutral and academic as possible, since discussions of political priorities and agendas tend to get heated. I'm not so arrogant as to assume that "my best" is the same as "fully successful", so I'm throwing this open to community wiki in the hopes that anyone who can tell the story better will revise.