Just as one cannot truly appreciate Shakespeare without understanding that the plays would be performed on stage and what an Elizabethan era playhouse would be like, one also cannot fully appreciate Homer without knowing how the Iliad or the Odyssey would be performed and what that performance would have been like to see and hear. The epics as we know them today have their roots in oral poetry going back to the late Minoan (or more specifically Mycenaean) period (about the 12th century B.C.E.) and took their extant form in about the 8th century B.C.E. in early pre-Classical Greece. Thus the Odyssey is from a culture separated from us today by about three thousand years. That is a gap in time comparable to that from the present day to a galaxy-spanning civilization envisioned in some space opera. One needs some genuine mental effort to imagine the context of the Homeric epics.
To a modern ear, ancient Greek even in daily usage would sound like it was being sung. (For language references, Goodwin’s Greek Grammar is an excellent thorough reference and Athenaze is a good basic reference.) The three accents of ancient Greek– acute, grave and circumflex– were tonal accents, not stress accents. In musical terms, a Greek speaker generally kept the pitch of his (or her) voice constant except on accented syllables. An acute accent indicates that the voice would go up half a step, returning to the basic pitch on the next syllable without gliding. A grave accent was similar except that the voice went down half a step instead. A circumflex accent marks where the speaker’s voice would on that syllable first go up half a step and then down half a step. This up and then down gliding of the voice is the practical reason a circumflex accent always lengthens a vowel. Moreover, the terms long and short in referring to vowels in ancient Greek are entirely literal; a long vowel (like eta η or omega ω) is held twice as long as a short vowel (like epsilon ε or omikron ο). The tempo would vary with emotion.
Poetry was so largely popular in the ancient world because it used and emphasized the musical qualities of Greek and most other ancient languages (including Latin). In modern terms, the concepts of poetry and music were inseparable in the ancient world. Music was built into the language.
The traveling bards who performed the Homeric epics were the musical pop stars of their day– complete with fans and followings. A full performance of either the Iliad or the Odyssey would be an all day event staged in the home of wealthy patron or in an amphitheater. A professor of mine compared the epics to rap music in the sense that they were to the ears of the listener spoken but yet at the same time distinctly music. One can hear a modern rendition of the Iliad and of the Odyssey respectively here and here; the tempo strikes me as overly slow but I think this is for the aid of understanding by non-native speakers.
Poetry abounded in the ancient world, as music which fits the comparable niche in popular culture does today. The works that people knew best and which the best of the best performed professionally were the Homeric epics. A performance of Homer by a bard held an appeal comparable in today’s terms to both a rock concert and a long stirring movie in full screen at a good cinema.