This depends on what your notion of "civilization" is. If you mean "civilization" as something like "nation" or "nationality," then the poem celebrates the Trojan War and the values that caused it, and, in so doing, it helps define Greek civilization. The historical accuracy of the poem in this sense means less than the artistry of it; the story of Hector and Achilles is a kind of artifact. By reading the poem, we learn something of what it meant to live in ancient Greece.

If, on the other hand, you mean "civilization" in the broader sense of being "civilized," or culturally sophisticated, the poem must be read in a different way. In this case, the barbarism of the battle scenes and the passions of individual characters—Agamemnon's lust for power, Achilles's pride, Hector's sense of duty—serve to humanize the events of the poem.

Because the poem dramatizes the internal conflicts these characters face while at the same time placing them in the context of the chaos of war and forcing them to make choices with real consequences, the story dramatizes what, for lack of a better term, one can call the "human condition." That is, because the poem humanizes these characters, as readers, we engage with their stories. In understanding the choices they make, we can evaluate our own choices and, with luck, make better ones. In short, the poem contributes to "civilization" by reminding us what it is to be human.

How has The Iliad influenced our understanding of civilization?

I think that one way Homer's work has contributed to our notion of civilization is through its storytelling merits.  Homer's story is a great one, replete with challenges from Gods, struggles between human beings, and ensuring that human problems are powerfully compelling and yet seen in an accurate context in terms of their impact on others.  Within this storytelling element arises some wonderfully rich characters.  Achilles as a brooding, yet skilled warrior is contracted with the honor- bound, yet sad Hector.  These two characters have helped to configure where human consciousness can lie and how tragedy can be seen as an inseparable part of the human experience.  The book is also powerful in reflecting how a society based off of war is not one to be glorified.  It is difficult to gauge if this was Homer's intent, but the work stands out as one of the most defiantly anti- war writings.  It reflects how societies that are based off of war are doomed to fail, and in doing so, almost repudiates the Classical standard of warfighting as being socially acceptable.