Well it didn't exactly happen "way back when", it occurred over the course of the late 1960s.

Traditionally the Democrats have been advocates of the working class and broad popular movements. The Republicans have always focused on big business and monetary policies. Prior to the Civil Rights act, Southern Democrats (called dixiecrats) reflected racist sentiments among the white working class. After Johnson passed the Civil Rights Act, the dixiecrats felt betrayed. Then in '68 and later in '72, Nixon began appealing to these white working class voters in the South through what his own campaign called "The Southern Strategy". This electoral strategy relied on what many call "Dog Whistle" rhetoric, with lots of language about Law and Order, and lamenting the dangers of "Civil unrest" to gain the support of essentially racist Southerners opposed to the Civil Rights movement. The Southern Strategy worked brilliantly for Nixon, and as time has gone on Republicans have relied on it more and more.

For their part, the Democratic party embraced the glowing legacy they received in the wake of the Civil Rights movement, even though people like LBJ had actually been dragged kicking and screaming to the side of the Civil Rights movement by activists. Starting with the Clinton presidency in particular, the Democrats began relying more and more on "Identity politics" to gain electoral support rather than the old-fashioned "class politics" of the past. So since the early '90s Democrats have relied on appealing to so-called "minority" groups such as African-Americans, Women, LGBT people, and Latin-Americans on the basis of their unique identity and issues (affirmative action, equal pay, reproductive rights,marriage equality, hate-crime protections, immigration reform); more than they have on appealing to working class people and labor unions (wages, collective bargaining, work place regulation, social safety nets); which had been the bread-and-butter of Democratic party strategy since the late 1800's.