At the time, Republicans were more socially liberal and Democrats were the social conservatives. This pretty much reversed during the 1950s and 60s with the emerging Civil Rights Movement. Democrats saw the potential voting power of African Americans as well as began moving into notions of equality and started courting African Americans.

That said, economically, the parties have pretty much stayed the same in terms of courting classes. Democrats since their emergence with Andrew Jackson have always been champions of the "working man." Republicans on the other hand, emerged as champions of capitalism and free enterprise. This notion actually had ties to countering slavery because many Northerners saw free labor as having an advantage in terms of production over slave labor (would you work harder if I actually paid you, or just so that I don't beat you?).

In the 1950s and 60s, there were many social conservatives in the South that were Democrats. As the party moved further and further to the left socially, these folks felt that the party was leaving them. Ultimately, those socially conservative Democrats became socially conservative Republicans. If you check through some of the Southern Democratic Congressmen at this time, a good number of them changed their party affiliation to Republican.

TL/DR the Democratic and Republican parties evolved and switched their platforms on social issues.