There are a couple of different ways of tackling this question and there's always more that can be said, but the heart of your answer lies in how people of the era thought about what it meant to be smart.

Generally speaking, the reason why men in positions of power (or with access to publication opportunities) were so interested in Greek and Roman authors was because the men around them were. As were the men they admired, the men who taught them, and the men they read. In other words, one of the the prevailing theories of knowledge was that being smart meant learning hard things and smart men knew Greek and Latin, so men wanting to be smart learned Greek and Latin.

In order to get into one of the Colonial Colleges (a previous question on the admission process), a young man had to know particular passages by particular authors - which meant his tutor or teachers had to spend a great deal of time helping him learn the authors' work and lives. As did the men who chaired admission meetings or acted as college presidents had themselves memorized texts by Greek and Roman authors, as did the men they admired or sought to emulate.

To put a finer point on it, knowledge was thought to have degrees of complexity. Greek and Latin were the foundation - meaning they were the hardest and most essential to learn. Math, modern languages, some sciences, and rhetoric helped fill out the "classical" curriculum. However, things like literature, especially poetry, and history were almost treated as frivolous. Men would spend time reading history and citing modern authors, but when it came time to show off their brain muscles, they did what the smart men they admired did and namechecked long dead Greek and Roman historical figures.