That's a pretty hard question to answer authoritatively because many different people have offered many different theories over the years - the debate about this has basically been going on since (or maybe even before) the first "hippies" appeared. Some of the most popular theories include:

• A growing dissatisfaction with American consumer culture among young people who had grown up during what many considered to be the golden age of American consumerism, the 1950s. According to this argument, many young people who had been raised in an era when most Americans assumed that money (and goods) could buy happiness were among the first people to reject that assumption, when they realized that in spite of being among the wealthiest and most goods-rich generations ever, they weren't happy. In response, they started to try to find a more "authentic" way of living, separate from and outside of consumer/capitalist culture, and which they believed was both more moral and more rewarding. The prototypical hippies were The Diggers (in San Francisco), and for them the rejection of consumer culture was paramount. (Note, though, that the Diggers were also artists, and they also drew on a long tradition of bohemian, avant garde rejection of mainstream culture).
• Young people becoming aware that the "American ideals" of freedom and fairness that they learned about in school (especially in Cold War-era civics and politics classes) weren't really being followed or lived up to by their own society, particularly in the context of race relations (the civil rights movement) and the Vietnam War. There's good evidence for this view in the founding document of Student's for a Democratic Society, the Port Huron Statement, which suggested that the young people of the 1960s were "looking uncomfortably to the world we inherit" and seeing a lot of "paradoxes" - between the rhetoric of the "land of the free" and the oppression of black people, for example, or "proclaimed peaceful intentions" of the United States and its militarism. (Note, though, that the people who founded SDS did so long before "hippies" were really a thing. They're a better example of the emergence of the "New Left" of the 1960s than of the emergence of 1960s counterculture, although there's a lot of crossover there).
• Young people rebelling against increasingly strict rules and regulations that were designed to control the conduct of students: during the post-war era, high school became a basically universal experience for teenagers in the US and the GI bill sent many veterans to university. That meant an increasingly diverse student population, and educators responded by putting more and more rules in place to try to make students behave the "right" way. In the 60s, students rebelled against that and asserted that they had a right to greater independence - most notably through student revolts like the Berkeley Free Speech Movement. But again, not everyone involved in that movement was a "hippie."
• Young people realizing that American society/culture was not going to offer them meaningful work that would make then happy - that it was basically an exploitive "system" that was designed to produce pliant workers and enrich corporations, rather than to actually make the most people happy or allow them to lead rewarding lives. This view was informed by the writings of people like Paul Riesman (The Lonely Crowd), William H. Whyte (The Organization Man) and Paul Goodman (Growing Up Absurd), all of whom had criticized the "mass society" and corporate/commercial structure of American life during the 1950s.
• Some people have argued that the post-war baby boomers were raised to become hippies, in that they had a more comfortable childhood than most previous generations and were encouraged to do/think/demand what they wanted by "permissive" parents. Social conservatives like James Dobson, in particular, have argued that this generation basically failed to ever "grow up" as a result, and that the whole hippie subculture was basically comparable to a child throwing a tantrum and not realizing that they can't have everything they want - or wanting all play and no work all the time.
• Many social scientists argued that the hippie subculture was the product of the rapid social and technological change of the post-war era. Margaret Mead, for example, believed that rapid technological change had made many of the cultural norms of adult Americans obsolete, that young people had realized this early on (because they were more at home with the new technologies) but that adults had not, and as a result the "generation gap" was a symptom of adults trying to impose/maintain a culture that younger people already knew was irrelevant to a changed world.

Those are just some of the "explanations" for the emergence of 1960s counterculture that have been offered over the years. There are many variations on each of these, and lots of others out there that I haven't mentioned here.

There's a couple of books about this that I'd recommend if you want to read more: Peter Braunstein's ImagineNation (that's an edited collection of essays about 1960s counterculture) and Terry Anderson's The Movement and the Sixties (which is about the 60s more generally, but also has a pretty good introduction to the emergence of 60s counterculture). iirc, David Farber's The Age of Great Dreams also has a pretty good introduction to the origins of the counterculture in it.

One thing I'd note, though, is that you should be very careful not to essentialize "1960s counterculture" and assume that it was just one thing; there were many countercultures (feminist, black power, drug, student, etc) and many people who we might think of as part of "the counterculture" today would have had serious disagreements in terms of what their goals were or why they were doing what they did. It also took very little time for "counterculture" to go mainstream and get commercialized, at which point it was really more of a fashion or a money-making enterprise than anything else.