Much better to read this after having read both of Homer's poems. Graziosi addresses the two biggest questions I had with these stories.
Regarding the Iliad: I was bewildered not to see any of the major plot points that everyone learns about the Trojan War (Judgement of Paris, Achilles's death, the Trojan Horse) really mentioned at all. Graziosi explains how these stories were captured in other poems that formed a broader cycle. And she answers the next obvious question for me, which was why it's the Iliad that has survived, if few people would tell you what happens in it, without having read it. Graziosi argues that the events of the Iliad mirror those of the broader Trojan War, while focusing on a single plot device, Achilles's wrath, to drive plot. Funny to argue the Iliad has an intricate (doll house) design, when there's so much blood and gore!
Regarding the Odyssey: I struggled getting through the latter half of the book, and in particular the ending, having grown up only knowing the adventure story of the first half. Graziosi cited no less an authority than Aristotle, who in his Poetics, does a much better job of explaining the Odyssey than our current culture. Aristotle emphasizes the latter half, and casts the epic poem as a story of Odysseus's domestic concerns. And it was a relief to hear even Aristotle found the ending (a deus ex machina where gods cause a collective amnesia across Ithaca so everyone forgets that Odysseus and Telemachus just killed a few dozen of the best residents of the island, who had been suitors for the throne in Odysseus's long absence).
Beyond complaining about the ending, Aristotle (and Plato) also debated who exactly Homer was, and what he wrote. How exciting to think that this author was as mysterious to them as he is to us today! It's a reminder of just how long history is, akin to points that we are nearer in time to Cleopatra today (2000 years difference) than she was to the Pyramids of Giza (2500 years difference).
Another fascinatingly foreign point about Homer is that Homeric Greek was never spoken by any actual community, but was a melange of various Greek dialects into one best suited for hexameter form. I suppose this is not uncommon in art, as nobody ever spoke like they do in the KJV / Tyndale Bible.
One last mystery solved about Homer for me.... I was used to hearing the repeated phrases ("wine dark sea", etc), and understood them to be devices to help the poem's recitors, but was surprised when seeing they aren't always exactly the same form (e.g. "luminous Achilles", "swift footed Achilles" or "swift footed, luminous Achilles"). Graziosi explains that these varied length names were just to fit the character's name into various slot lengths needed to complete the line's hexameter. Makes sense!