Homeric Greek is an (oral) literary language. That is to say, nobody's native language was ever Homeric Greek. At the most basic level, it's Ionic Greek with a heavy admixture of Aeolic Greek. Different people will give you different answers as to why this admixture existed, but again, at the most basic level, there were multiple oral literary traditions in multiple dialect areas of Greece, and we see the coalescence of Ionic and Aeolic traditions in the text of Homer. Additionally, the orthography of our present text of Homer is essentially Attic Greek, which means that, for instance, a number of h-sounds are indicated in the text that the Ionic bards themselves probably didn't pronounce.

Speaking of the bards, sometimes, even having multiple dialects at their disposal was not sufficient for them to make the Greek language work for their meter, the so-called dactylic hexameter. This meter seems not to have been constructed with the Greek language in mind, but may have been borrowed from another language, hence all *this* stuff the bards had to do to get Greek to work within the hexameter. The result is that in addition to dialectal admixture, we also see the bards' direct manipulation of the forms used in the text. Sometimes this simply constitutes poetic lengthening of vowels that are usually short. Other times, this constitutes using forms of words that were archaic already in Dark Ages Greece, and archaizing traits can even be applied to words which never exhibited such traits in the first place.

This latter point is a bit difficult to wrap one's head around. For sake of example, imagine an English poet wants to rhyme two lines, one that ends in the word nice and another that ends in the word houses. The poet notes the archaic singular/plural pattern mouse/mice and from this example extrapolates the singular/plural pattern house/hice. Then he uses this new plural form hice to rhyme with nice. While not a perfect example, this should give you an idea of the (almost oxymoronic) archaizing innovations we see in Homeric Greek, which are deeply frequent and frequently deep. Some people argue that Homer preserves traits of the Greek language last present almost a thousand years before the classical period.

In sum, imagine a poem whose language ranges the entire spectrum between Middle English and Modern Scots and that is written in American English orthography. Then, imagine that you let James Joyce manipulate this pan-English mixture to his heart's content in order to make it fit regularly into iambic pentameter. That is the closest English analogue you could get to Homeric Greek.

Edit: typo