The Homeric poems seem to have been composed in the early-to-mid-7th century BCE, ca. 670-650 in the case of the Iliad, but though the stories of the Trojan War and of Odysseus' return were popular throughout the Greek and Etruscan worlds from a very early date, the poems themselves seem to have had only limited dissemination for the first 100-150 years of their existence. From about 550-520 BCE onwards, they experienced a meteoric rise in popularity, and they have never lost that popularity since.
We first hear of Homer in a historical context in Sikyon, ca. 570 BCE, when the local king allegedly banned performances of Homer because his poetry praised his rivals the Argives. However, that story only makes sense if we understand it as talking about the lost Thebaid, not the two epics that have actually survived. An even more doubtful reference tells us that the 7th century BCE poet Kallinos attributed the Thebaid to Homer. These stories are the only references to Homer or to Homeric poetry that we can be confident of prior to 550 BCE: we have no evidence of anyone knowing the Iliad and Odyssey at all.^1
After 550, everything changes. Previously, vase-paintings of Trojan War scenes used material from throughout the legend of the war; after 550 there is a sudden, very sharp increase in popularity of scenes relating to the Iliad, and by the early 400s it's possible to see direct influence from the Iliad in the pictorial tradition. Stesichoros, Herakleitos, the Hymn to Apollo, and Simonides making explicit references to Homer or "the man from Chios". Theagenes of Rhegion writes the first piece of scholarship on Homer, giving an allegorical interpretation of the gods. A fairly compelling argument has been made that the Iliad was first popularised at the Panathenaia of 523/522 BCE; the argument is conjectural in parts, but we do have good evidence that Homer wasn't performed in Syracuse until 504, so it's certainly not implausible. And finally, any kind of fixation of the Homeric epics in writing relies on writing having a particular cultural function, and Jesper Svenbro has shown from epigraphic evidence that it was only ca. 540 that written texts began to acquire the function of reproducing an utterance, as opposed to being an utterance (that is: earlier inscriptions refer to the inscribed object as "I", or address the reader as "you"; writing was supposed to be an utterance that made sense at the moment of reading, not an archive of an old utterance).
What seems to have happened is that up to this point, the Iliad and Odyssey were transmitted orally in a poetic tradition on Chios, as the heritage of a group of poets there who were called (or who called themselves) the Homeridai, "sons of Homer". This appears to be why Simonides and the Hymn to Apollo treat "Homer" as Chian. The name "Homer" itself was a marker of a poetic heritage; whether or not a real man of that name had ever existed, that was the meaning it had for people of the 6th century BCE -- a label for a body of poetry linked by the affiliation of the poets who created it: the Iliad and Odyssey, but also the Thebaid, the Epigonoi, the Hymn to Apollo, the Margites, and other pieces. Similar heritage labels emerged elsewhere, under the names of "Orpheus", "Mousaios", "Hesiod", and possibly others (I suspect Sappho and Solon should also be regarded more as heritage labels than as historical individuals, even if those individuals actually existed).
After 550-520, the Iliad and Odyssey were both treated as classics, memorised by the elite, recited at aristocratic parties, performed in major civic competitions, imitated and excerpted by other poets, quoted by philosophers, and studied by scholars. From 520 BCE up to the present, their status as the preeminent classics of Greek literature has been unchallenged. In western Europe they were much less read in the period ca. 300 CE-1500 CE, but before and after that date-range they have also reigned there as the key foundational works of Western literature.
Note
^(1)There are several other pre-550 BCE references that have been interpreted as references to Homer, or as quotations, but since the nature of Homeric poetry and the mythical content of the poems is in every way traditional, it is more parsimonious to interpret these as cognate with Iliadic/Odyssean material, rather than derived from Iliadic/Odyssean material. These references are: (1) "Nestor's cup", ca. 730 BCE; (2) Tyrtaios fr. 10.21-30 West; (3) Alkman fr. 80 Page; (4) Mimnermos fr. 2.1-4, fr. 14.1-3; (5) Mousaios fr. 5 D-K; (6) Alkaios fr. 44 and fr. 395; (7) the Hesiodic Shield as a supposed imitation of Iliad 18; (8) knowledge of the Iliad attributed to Solon (Plutarch Solon 10, Strabo 9.1.10).
For these works to have survived for so long, it must be assumed that they were popular, but in a different sense than you might think. To understand their popularity we have to note two things: First that it is accepted that these works were originally told orally, and were later written down into more or less the form we have today. Second is that for the works to survive to today they must have been written down several times in several places.
The First Part: This is called Oral-formulaic Composition and was first noted by Milman Parry and Albert Lord in what would later be called the Parry-Lord Theory. This theory essentially stated that epics were NOT originally written down, but instead passed down through oral tradition, particularly through bards. These bards remembered the key plot points of the stories (e.g. Achilles killing hector, or Patroclus' aristeia), and then filled in the rest of the details as they sang. If you read certain translations of the works, you will notice how they could do this. Things like "swift-footed Achilles" or "Enduring Odysseus" are repeated often and are examples of how the bards might sing the song while maintaining meter and reminding the audience of what the characters were like. This is just an example of the legacy of the oral tradition remaining in Homer's work.
Typically the bard would sing these songs over the course of a few days. You will notice the Iliad is 24 books long, with climactic moments in the 8th and 16th book (The Trojans pushing back, and the death of Patroclus) which would suggest the story was told over three days, so that people would be excited to return the next day, making the bard more money.
Now what does this mean? There were likely many tales sung by bards before writing even existed. For these two tales to be chosen to be written down is significant in itself. Ancient writing took a long, they didn't have the printing press or any other machine to mass produce these works, so they must have been important for anyone to even bother writing them down. On top of this, the fact that they survived until today (or the Renaissance to be exact) is even more significant.
In short, a huge amount of the written works of the world were lost from the time these stories were first written (~760 B.C.) until today. So for us to have essentially complete editions of these works, on top of the fact that they wrote them down at all, is incredibly significant, and would indicate their popularity to the Greeks and the scholars who followed them (of every age).
This is my first post here, and a lot of this comes from my own knowledge through classes, I included a few sources but if you would like more let me know.
Edits for grammar.
Sources: A summary of the original work: Parry, Adam (editor) (1971), The making of Homeric verse. The collected papers of Milman Parry, Oxford: Clarendon Press
A more recent analysis of the Oral Tradition: http://www.yale.edu/heyzeus/spring2004/zeus_sarah_price.pdf
Also a former professor of mine was the son of Albert Lord, I took two classes from him, one on epics, and one on Tolkien.
hi! you may be interested in these posts
how commonly known would the story of the Iliad, have been in the second century BC classical world?
of possible tangential interest