Like all ancient Greek literature that has survived, the Odyssey was transmitted via manuscripts that were kept and copied within the Classical Greek world, then the Hellenistic world, then the eastern Roman empire (and later Byzantine) world. Its survival depended on different things depending on the period we're talking about.

We emphatically do not know that it was transmitted for centuries before it was written down. We know it was written down by the 520s BCE; many (most) scholars imagine it was written down in the 600s or even the 700s, though the reasoning for pushing the date early is not compelling. It was probably composed in more or less its final form by the 650s BCE.

There are no strong reasons to imagine its origins are much older than the 600s BCE. Linguistically, we don't have any rigorous data to suggest how long it should have taken the formulaic style to evolve: there are a handful of elements of Homeric formulae of Bronze Age date, but only a tiny, tiny minority. Some estimates put the development of the Ionic formulaic system as taking place over a century or so. The Trojan War legend is most likely to have evolved in the context of Greek colonisation of the Troad in the 700s; the sea travels depicted in the Odyssey are clearly the product of the period when Greeks were colonising beyond the Aegean, starting around 800 BCE and continuing into the 600s.

By the late 500s, and in the 400s, it existed in written copies, thanks in part to the popularity of the Homeric epics at the Panathenaia festival in Athens. Homer's big break-out moment at the Panathenaia seems to have taken place around 523/2 BCE, maybe plus or minus a few years: that's a year after Kynaithos put on a compiled Hymn to Apollo performance at Delos, adopting the persona of Homer.

We imagine that official Athenian copies were one way that the Odyssey was preserved to the Hellenistic period, but there were several other veins as well. We know figures like Aristotle had good copies; but we know much less about their origins. Some scholars imagine that they were rhapsodic copies, put on papyrus by people recording "unofficial" performances by rhapsodes. They agree broadly with the received text, though with divergences that are wild by the standards of classical philologists (the divergences are very mild by the standards of mediaeval poetry).

After this point, state policy ceased to be so important: scholarship took over. In the 3rd century BCE, centres of scholarship made a concerted effort at making standardised versions of earlier literature. They compared different copies, and began to develop detailed critical methodologies for working out how to reconcile them. At this stage the scholars' interest was not so much in recovering the history of the text, as in reconstituting the most perfect text, as they saw it. For that reason their work has methodological flaws. But they weren't irresponsible: we have detailed textual notes in some copies of Homer that tell us what scholars of that period thought of the variants that were to be found in different copies, and they do record the differences, not just throw away the ones that they don't like. Two key 3rd century figures in this phase are Zenodotos and Aristarchos; in the 1st century BCE, Didymos is another.

We don't know exactly which copies were the most authoritative for the transmission of Homer in late antiquity. Aristarchos was the most revered scholar on Homer's text in antiquity, but surviving mediaeval copies certainly don't follow his decisions unquestioningly. Didymos' text may have been more influential. But the Iliad and Odyssey were so popular that there was never any doubt that they would be among the texts that would survive: in the papyrus fragments that have been found from Roman-era Egypt, the Homeric epics are by far the most popular literary works.

Copying continued; scholarship continued; Homer continued to be part of the curriculum in Byzantine schools and higher education. We have a bunch of treatises from the 9th-12th centuries on Homer, including some detailed commentaries with information about the history of the text that survives nowhere else. The earliest complete copy of the Homeric epics dates to the 10th century, by which time these treatises were being written.

We do have plenty of earlier fragmentary copies too: some papyrus copies from the 3rd-4th centuries cover large amounts of the text, and the very earliest papyrus fragments date to the 3rd century BCE. There's a 5th century BCE potsherd that has one line of the Odyssey written on it; a 4th century BCE religious tract quotes another line; a bunch more lines are quoted on vases from the 4th-3rd centuries BCE. So there's no real doubt that the text as we have it today is pretty close to how it was in that period. There were plenty of variations in the text in that period, and doubtless our text doesn't reflect any one copy of that period perfectly, but it's as close as makes no odds.

The biggest differences between our text and the text when it was first written down aren't to do with wording, but spelling. We don't know what spelling conventions were used in the earliest copies. The Hellenistic Greek of the 3rd century BCE had very specific standardised spelling conventions, and earlier poetry may well have been treated pretty violently when it was converted to those conventions. The debate on that is ongoing.