We may never know why the Greeks stopped believing in their gods because majority of the sophist writings—which probably explain the disenchantment of the Achean worldview—were burned.
Shortly after the events of 314 BC, the Greeks, under the influence of Spartan rule and guided by the philosophical truth-system of Plato, rewrote the past to fit the present and expunged the relativistic literature of those golden age political consultants and educators: the sophists.
It seems the Spartans couldn’t tolerate the the free market of ideas (religious and political) that populated the Greek ethos. It was annoying to continually defend their legal positions against experts at rhetoric, perception, and reason. Thus arose Plato’s idea of Absolute Truth. It was employed as the most effective antidote to sophist-driven relativism. A belief in Absolute Truth, of course, requires certainty and certainty nearly always tends toward violence and so was lost the great legacy of thought produced by early Greek thinkers. Most of the sophists, including Socrates, were killed for being independent thinkers.
I happen to think the greatest attribute to our modern scientific thinking (thanks to Francis Bacon) is not the preponderance of well documented evidence organized into air-tight theories of how the world works, but the discipline of doubt and skilled skepticism. Science, in my view, is chiefly a habit of mind and not simply a body of facts organized by parsimonious theories. Science is the disposition of systematically learning for oneself about the material and social world around you—our best and most ancient use of mind.
Hopefully that is a historical clue that will offer some ideas as to why the Greeks stopped believing in their gods.