Why? Fact-free speculation is what blogging excels at, so here's my take. Most of the major parties in today's western democracies were fully formed by the middle of the 20th century, during a time before the current culture wars were a twinkling in anyone's eyes. So regardless of how and why they were originally founded (the Republican Party, for example, was originally associated with the anti-slavery movement), by the 1950s they were primarily associated with purely economic positions. They either represented the working class or else they represented big business and the rich.
So the question is: when the 50s and 60s dawned, why is it that it was mostly the economically liberal parties that supported the emerging social revolution? This strikes me as a pretty easy question to answer: it's because the founding principle of most liberal parties was economic egalitarianism. When the postwar era rolled around and the current crop of social issues became important, it was only natural that economic egalitarianism morphed into social egalitarianism, and that in turn led to support for civil rights, feminism, and gay rights. Support for things like affirmative action, criminal justice reforms, and abortion rights were obvious corollaries.
This doesn't explain everything. Nothing explains everything, after all. Environmentalism, for example, is something that I suspect everyone naturally supports unless they have some reason not to, and the main reason not to is that it interferes with business interests. So opposition to environmentalism comes mostly from conservative, pro-business parties, while everyone else supports it. It has nothing much to do with egalitarianism.
Ditto for some other social issues, like gun control and school prayer, which are slightly mysterious. They might be associated with the urban bias of liberal parties, or they might just be an artifact of tribalism. After all, once you've drunk enough of the Kool-Aid on either side, you tend to drink the rest.
But a commitment to egalitarianism probably explains most of it. If you're committed to breaking the stranglehold of the ruling classes, that just naturally leads you down certain roads. Likewise, if you think the current hierarchy works pretty well, that leads you down other roads. Economic justice and social justice are two sides of the same coin.
POSTSCRIPT: Is egalitarianism the underlying principle that guides everyone's political beliefs? Of course not. Libertarians, for example, just want government to leave them alone, which leads them naturally to support social liberalism and economic conservatism. The problem is that human beings are social creatures, so "leave me alone" has never attracted a huge following as a guiding principle.
As for the rural populist types (socially conservative/economically liberal), I'm not really sure wat motivates them. Whatever it is, though, it seems to have a relatively limited base these days.