A lot of schools don't give students "The Odyssey" and other ancient epics to read until they're near high school level (at least the unaltered translated versions). This assumes that the texts are at least somewhat difficult to read, but were they difficult to understand for literate Greeks at the time of release? Any sources to your information would be much appreciated.
Whether they're difficult or not depends to a large extent on which translation you're using. If schools are using the antiquated Butcher and Lang translation --
Then the lord Aegyptus spake among them first; bowed was he with age, and skilled in things past number. Now for this reason he spake that his dear son, the warrior Antiphus, had gone in the hollow ships to Ilios of the goodly steeds...
then of course it's going to be harder than for students reading, say, Lombardo:
First to speak was the hero Aegyptius,
A man bowed with age and wise beyond telling.
His son, Antiphus, had gone off to Troy
In the ships with Odysseus...
That said, the Iliad is definitely for an older audience than the Odyssey. I'd happily recommend the Odyssey for almost any age, as it's simply a ripping read -- a ripping read that happens to have almost limitless depths and subtleties, but still a basically good story. The Iliad, by contrast, has some definite stumbling-blocks: the lurid death-scenes, the speeches, and of course the Catalogue of Ships.
To get back to your question: ancient Greeks didn't have the concern of which translation to use, of course. I'm not aware that we've got any evidence on how "readable" they found the epics in terms of their content. But they certainly found the language difficult.
Some of the earliest extant scholarship from antiquity is glosses to the Homeric poems, explaining the meanings of the old-fashioned words for a contemporary audience. Here "contemporary" means the late 5th century; for that is how old some of the D scholia are. The difficulty was only partly because of age: it's also because the poems are in a weird mix of dialects, dominated by Ionic, along with a substantial dose of Aeolic and a smattering of Attic and Arcado-Cypriot, and even a handful of formulae pre-dating the Mycenaean era. There are also many linguistic quirks that Classical-era Greeks wouldn't have noticed, but which are responsible for a wide range of irregularities in the poetic construction. Yes, they found the style hard going.
It sounds like you are unaware that during the "time of release" of the Iliad it was an oral epic poem and not written down at all. So no one would have read it until it was written down centuries later. You should probably clarify if you are wondering about how well the oral poem was understood or if you are wondering about the reading comprehension centuries later when it was written.