It has nothing to do with ancient Greek culture or neoclassicism. Almost all American Greek-letter societies (including general/social fraternities, service fraternities, honor societies, and professional fraternities) follow the example of Phi Beta Kappa.
Remember that the traditional university curriculum included a heavy emphasis on classical texts, and the tiny elite who enjoyed a post-secondary education for most of modern history would have had many years of schooling in Latin, ancient Greek, and Hebrew.
There had been student societies such as the FHC, established in 1750 at the College of William and Mary. The letters FHC were the initials of a Latin motto, Fraternitas, Humanitas, et Cognitio ("brotherhood, humanity, and understanding"). As the motto was secret, the FHC was popularly known as the "Flat Hat Club."
In 1776, a student named John Heath was denied membership in the FHC, so he formed his own organization in response. Instead of a Latin secret motto, he used a Greek one: Φιλοσοφία Βίου Κυβερνήτης ("love of learning is the guide of life"), and so the organization became known as Phi Beta Kappa.
Phi Beta Kappa was eventually forced to become an open society, but all collegiate Greek-letter societies more or less follow on the model of these early student clubs. Some, like Farmhouse or Triangle, do not use Greek letters; others, like Delta Upsilon, are open societies rather than secret societies, but for the most part they take the same model: letters representing a secret motto or principles; a creed or other philosophical basis for organization; and public and private tests for identifying members, such as a badge or secret handclasp.
Incidentally, while a certain stereotype of social fraternity members as athletes is reinforced in postwar films and television, there is no institutional basis for that association. The first "social" fraternity, Kappa Alpha Society (est. 1825 at Union College), grew out of a literary society, and the "social" purpose of the group was to groom its members to contribute to society, not merely to socialize.
Baird's Manual of American College Fraternities is the canonical reference on the fraternity and sorority system.