I can't give a thoroughly historical answer, but the Republican Party was indeed founded as a radical Abolitionist, that is, anti-slavery, party. In the period of Reconstruction after the Civil War, Republican federal leadership was responsible for introducing and enforcing the 14th Amendment and supporting legislation, which was the beginning of the notions of "racial equality" which is today considered a left-wing position. Meanwhile, the Democrats were still strong in the South and held a conservative position that led to voting suppression and other forms of discrimination against newly free African-American citizens. The Compromise of 1877 that ended the Reconstruction period gave white Democrats in the South a largely free hand to implement so-called "Jim Crow laws" that blocked black citizens from the rights guaranteed by the 14th Amendment. It's definitely easy to call Republicans "socially left-wing" and Democrats "socially right-wing" in the 19th century. Even in the 20th century, in terms of economics and the role of government, the two parties were very different than they are today: Republican President Teddie Roosevelt was militarily jingoistic for the time period, but he was also famously for government economic intervention in terms of "trust-busting", i.e. regulating markets by legally breaking up monopolies. There are a lot of things like there where the parties don't map 1:1 to their modern policies.

The "flip" that people talk about was a gradual process that took several decades but could in general be thought of as starting in 1933 with the inauguration of Franklin Delano Roosevelt and ending in 1968 with the election of Richard Nixon. FDR is credited as a "modern, left-wing" Democrat for economic reasons; in an effort (which was debatably successful depending on who you ask) to mitigate the economic effects of the Great Depression, FDR engaged in Keynesian Economics, rapidly expanding government bureaucracy and pouring federal funds into subsidizing and employing Americans deeply affected by the Depression in his New Deal programs - something that defines modern Democratic economic policy and is anathema to both modern "small government" Republicans and also his contemporary Republicans, who feared the specter of Socialism in such heavy government investment. Republicans thus began to cast themselves as the party of small government and anti-Communism, leading directly (but mainly after the end of the Depression and World War II) to radicals like Joe McCarthy and Roy Cohn becoming representative of the party in the public eye and a slightly more moderate anti-Soviet jingoism becoming the policy for Republican Presidents and Presidential Candidates through the end of the Cold War.

Regardless, Republicans in this era became deeply concerned with "socially conservative" viewpoints as the Cold War was kicked off in earnest, seeing the promotion of Christianity, free-market capitalism, and other things we associate with the modern party as a way to differentiate and resist the world Communist revolution. That said, there was definite waffling on it - Republican President Eisenhower, for example, was an idealistic and stubborn promoter of Civil Rights, and also notably suspicious of the Military-Industrial Complex, as he named it, despite being an incredibly famous general himself...but at the same time, his administration barred huge numbers of homosexuals and suspected homosexuals from the military. It wasn't an instantaneous switch, no matter how you look at it, but it's also notable that his Democratic successors Kennedy and Johnson kept up the same pressure for desegregation, voting rights, and support of the Civil Rights movement in general: something that greatly angered and arguably disenfranchised the Democrats' ancient base in the South. That's where Richard Nixon's Southern Strategy came in: by making his campaign contingent specifically on promoting what we would call "socially right-wing" values, he was able to court the former Democrats who were angered by the success of the Civil Rights movement and the government interventionism of the modern Democratic party and win two *landslide* presidential victories and essentially delineate the two parties as we know them today.

tl;dr it wasn't an instantaneous flip, but a gradual exchange of values and voting bases over a period of 25-30 years and part of an ongoing and evolving notion of issues and definitions in the United States that has been ongoing since the Declaration of Independence and continues today. You can't shoehorn everything into "left-wing" and "right-wing" boxes, but there was a definite exchange in the middle of the 20th century.