When and why did the American Republican Party become widely supported in the South?

Question

The American Democrats and Republicans are often labelled with the colours blue and red respectively. However, the historical "blue states" and "red states" were once inverted from their current meanings.

In the 1860 presidential elections (in which Abraham Lincoln was elected US president), it can clearly be seen that Lincoln's Republican Party won the majority of the northern states, while Douglas and Breckinridge (Democrats) were mainly supported by the Southern states.

In 1916, when Woodrow Wilson (a Democrat) was elected for his second term, support still came mostly from the South, while the majority of the Northern states voted for the Republican Charles Hughes.

However, the traditional Red-Blue state split became less obvious in the middle 1900s, with elections during this period showing less obvious north-south splits. At the end of the century, in the 1992 elections, the contemporary pattern of North-Democrat and South-Republican became obvious in the Clinton vs Bush elections:

The most recent election (2012) of Barack Obama (Democrat) vs Mitt Romney (Republican) showed an almost 100% flip from the 1860 states map, with Obama winning almost all of the Northern states, and Romney winning almost all of the Southern states.

When did this switchover of party allegiance between states occur, and why did it happen?

Answer 1

This started around 1964. Barry Goldwater (Republican) did better in the deep South than the rest of the country. George Wallace (third party) won most of those states in 1968. Richard Nixon (Republican) won them in 1972. Jimmy Carter was the last Democrat to sweep the South in 1976. Ronald Reagan and George H. W. Bush (both Republican) swept the South for three elections, with the single exception of Georgia (Carter's home state) in 1980. The Southern ticket of Democrats Bill Clinton and Al Gore split the South for two elections. Texan Ross Perot's presence on the ballot may have acted as a spoiler.

Starting with 2000, the Republicans have taken every Southern state in each election, with the exception of North Carolina in 2008. Florida went for Barack Obama in 2008 and 2012, but I'm not counting it nor Virginia as Southern states. The issues there are somewhat different.

Why it occurred is more complicated. Part of this is that Southern states are more conservative than Northern states and the Republicans have become the more conservative party. As late as 1932, the Democrats ran on a more conservative, balanced budget platform. As late as 1964, by percentages, more Republicans voted for the Voting Rights Act than Democrats due to overwhelming opposition among Southern Democrats.

Part of this is outreach. Republicans tried to recruit Southern candidates while Democrats tried to recruit Northern candidates. The Republicans had to work harder to convince suspicious Southerners who still felt burnt from the Civil War. The Democrats actually had an easier time convincing blacks that they were no longer the party of slavery. Perhaps the Great Depression did most of the convincing for them. In both cases, once they switched, they committed about thirty years later. That's 1932 and 1964 for black voters and 1964 and 1994 for Southern whites. That realignment has only increased its polarization in the 2006-2014 period. Some of the last conservative Southern Democrats retired or lost a bid for reelection in 2010 and 2014.

As already noted in the comments, this question discusses the racial aspect of this in more detail.

Answer 2

@Brythan answered the "when", but the "why" was a calculated strategic political decision by the Republican party to appeal to (or perhaps stoke) racist sentiments among Whites against African Americans, especially in the Southern states. This is known as

The Southern Strategy

Quoting from the Wikipedia article:

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 to the Republican Party that had traditionally supported the Democratic Party. It also helped push the Republican Party much more to the right.

Despite never having quite given this strategy up, the party has actually sort-of apologized for it:

In 2005, Republican National Committee chairman Ken Mehlman formally apologized to the National Association for the Advancement of Colored People (NAACP), a national civil rights organization, for exploiting racial polarization to win elections and ignoring the black vote.