I'm currently taking an attic greek course in college right now and I began to wonder about this and also a followup question; if attic greek speakers recognized the differences between the homeric greek of the iliad and the attic greek that they spoke, would they also be aware of the fact that languages can change over time?
As a heads-up, if you don't receive a satisfactory answer to your questions: we do have Dr. Eleanor Dickey coming in tomorrow to do an AMA on language learning and education in ancient Greece and Rome. You may wish to ask her there!
It is worth noting that the Greeks would not generally have read Homer. The Iliad and Odyssey were performative poems, with extracts spoken out loud to audiences. When I'm teaching I sometimes use an admittedly slightly crude but I think helpful comparison: when we read Shakespeare his vocabulary and syntax can be unusual and difficult for us to follow until we get used to it. But when you see Shakespeare on the stage it is much easier to follow the flow of the language and the story.
As for the follow up question, there was certainly an awareness among the Greeks about how language could change and evolve. A number of Greek writers discuss the origins and development of individual words, and the acknowledgement of different dialects (such as the Attic you are studying) reflected their awareness of the way in which population movement and interaction change language, particularly I would say with regard to the Doric dialect of the Peloponnese and western Greece, which was understood to have been influenced by an influx of Dorians from the north in the Early Iron Age.
(NB: The "Dorian Invasion" was a long held tradition that is now heavily disputed. There is essentially no archaeological evidence for it. However, as far as your question is concerned, the fact that it was used by the Ancient Greeks as an explanation for the Doric dialect, does indicate at least that they had an understanding of the evolution of language.)
I hope that helped. Enjoy the course, καλή σου τύχη!