First of all, we cannot say with absolute certainty that the GI Bill caused the counterculture. It is much more accurate to say that it may have been one of the factors that contributed to the rise of the counterculture. Saying for certain that it was the cause of the counterculture is going a bit far.
That said, the basic argument for this point of view is that the GI Bill led to the rise of a prosperous, suburban society that left youth relatively wealthy but also dissatisfied. The GI Bill helped make the US both wealthier and more suburbanized. It helped to create wealth mainly by allowing ex-military personnel to go to college or to get other sorts of training. This allowed them to aspire to better jobs than might otherwise have been available to them. It helped to suburbanize America by providing home loans as well as by making ex-GI’s better off financially.
By doing this, the GI Bill (we can argue) created 1950s and early 1960s youth culture. It created a class of young people who had more money than ever before but who felt stifled by what they saw as boring, materialistic, soulless suburban life. This simultaneous wealth and disaffection helped lead to the rise of the counterculture.