To me, the main things that allowed this were:
- Prosperity. During the '50s and early '60s the US got very rich. When it did, that made people willing to pay more in the way of taxes for things like Medicare and welfare programs (economic liberalism).
- World War II. The war made people much more likely to accept the idea of an activist government. The government had run the whole war, and then it had isntituted the GI Bill -- things like this made people much more likely to believe in the idea that the government should be more involved in the economy.
- The backlash against the '50s ethos pushed cultural liberalism in the later '60s.
This is a topic that could encompass a book or two, or three. Be ready for amassing different and unique answers to this question. I think that some basics need to arise from this topic area. One particular element which has to be accounted for when examining the rise of American liberalism would have to be that the ending of the Second World War created a world outlook that featured two paradigms. The capitalism driven market economy with its political representation of democracy in America countered the Russian state sponsored Communist vision. The American conception of liberalism began to emerge set against this backdrop. The 1950's seemed to be a period that asserted this American vision to its maximum level, complete with high level paranoia of the Russians. When Sputnik was launched, the perception was that "all was not well with America," only highlighting how the conservative vision favored American supremacy.
At the same time, there was an emerging growth of expanding dialogue and seeking to change what is into what could be. It is here where American liberalism seemed to gather its greatest forces. The Civil Rights Movement which started in the late 1950s, demanding a transformation of what is to what can be, helped to feed it. Student activism at college campuses continued this with taking up the cause of Civil Rights for both people of color and women, as well as an examination of the class system which was enhanced by the 1950's consumerist and mass capitalist society. Additionally, college students began to adopt the belief that American advocated capitalism and democracy was merely code for imperialist practices which benefited the few over the costs of the many. This perception was tragically enhanced with the escalation of the Vietnam War, seen as a bulwark of American conservative values. The liberalist rise was fed by the opposition to the war and its expanded effort. The conservatively established society which preached an upholding of the Status Quo through the 1950s met its liberalized and transformative match in the 1960s.
This conflict was felt in both political and social orders, in music and fashion realms, and even in the domain of professional sports. The 1968 Super Bowl was the first to include the upstart American Football League (AFL), represented by Joe Namath and the New York Jets, against the establishment Baltimore Colts and Johnny Unitas. The long and unkempt locks of Joe Namath's hair stood in stark contrast to the buzz cut of Unitas in a Super Bowl which seemed to be more than just a football contest, but a competition between American forces of conservativism and liberalism.