As with all Greek literary poetry from that period, we rely on copies of copies of copies, in an unbroken chain from around the 400s BCE until the early Modern period. The oldest complete copy of the Works and days dates to the 900s CE, a manuscript held at the Bibliothèque nationale in Paris (Paris. graec. 2771); there are nearly 300 other more recent manuscripts.

We do have numerous fragments of older copies as well: an independent version cited by a Byzantine dictionary (the Etymologica); and about sixty ancient papyrus fragments found at archaeological sites in Egypt, mostly dating to the 100s-200s CE.

Based on all of these, modern editors reconstruct the oldest form of the text possible, and that's what you get in a modern critical edition (annotated with variants). This is very close to the text as it would have circulated in classical Greece, around the 400s-300s BCE.

As an alternate version of how transmission worked, here's a snippet from a response I wrote earlier this year, relating to Herodotus:

Ancient books were transmitted by an ongoing tradition of copying and re-publishing books. In antiquity this was done by booksellers who got scribes to make new copies; there was a severe decline in book culture in the Greek-speaking parts of the Mediterranean between the 6th and 9th centuries, and when it revived starting in the mid-800s, it was thanks to a mix of secular and ecclesiastical scholarship. Book culture in Constantinople was more or less continuous from then until 1453.

After that point, the text of Herodotus survived in copies that had mostly been previously transported to Italy, and disseminated from there. Manuscripts made in that era -- after the development of the printing press -- are often very important too. In the west, different manuscript collections had different levels of stability and permanence. For example, one collection now held at the Vatican, the Palatine collection, came from a collection that was captured in the Thirty Years War from the Electoral Palatinate in Heidelberg in the 1500s-1600s. This in turn had been swiped from the monastery of Lorsch when the monasteries were dissolved in the 1560s. And Lorsch in turn got its collection from a range of sources.

And here's a longer piece I wrote last year that gave more context.

So that's the story of transmission from the 400s BCE onwards. The transmission of the text before that date is less clear.

For my money, oral transmission must have played a key role. We have no good evidence that any written copies of any Greek literary work existed prior to the 540s-520s BCE; and some reason to think that written copies only began to be made around that date. If so, then all older poetry that has survived must have been transmitted orally for at least a century. We do have plenty of good evidence from the 400s that older poetry was memorised verbatim as a standard part of elite education, so this is a plausible picture. In that oral period, the main scenario for performance (and therefore transmission) is called the symposion, that is, social gatherings of elite men and enslaved women that served as a venue for social bonding between elite males, eating, drinking, poetry, sex, and acculturation of adolescent boys.

Then again, many scholars maintain that literature can only exist in written form; or that grave inscriptions a few lines long on stone and ceramics are evidence of literary poetry of hundreds of lines written on papyrus; and therefore all surviving poetry must have existed in written form the moment it was composed. This is an open question. As things stand there's no evidence either way, so it's entirely a matter of preconceptions -- how plausible one finds it that significant quantities of literary poetry were transmitted orally; or how plausible one finds it that manuscripts existed without clear evidence of their existence.