First, what is a Muse?
A Muse is a spirit of inspiration. The Ancient Greeks of the Bronze Age believed that one of nine Muses inspired different forms of prose, poetry, drama, rhetoric, or general writing. Thus, the invocation was a homage and supplication to these deities in hopes of best capturing the topic of the Epic poem, expressing the purpose of the narrative, but also incorporating mores and traditions familiar to that specific audience. Moreover, this invocation elicits one of the major themes of Homer's Epic poem: man and his relationship to a higher power.
Second, who is this Homer?
Homer was a blind story teller who traveled from town to town reciting the great deeds of legendary heroes of the time. Remember, roughly 700BC did not have the common contrivances that we are privy to today: no television, psp, ipod, iPhone, blackberries, tweets, etc. Homer was the entertainment of the time as he regaled citizens of great warriors like Achilles in "The Illiad" and model Greek men like Odysseus in "The Odyssey."In both Epic poems, Homer reveals the need to respect higher powers.
Finally, why does he need that invocation?
It seems ridiculous that a blind story teller is able to recite a story so complicated and so lengthy. In short, Homer and "The Odyssey" maintain the oral tradition. Through repetitive recitation, someone could hear and hopefully retain the story enough to recite it to another audience. This invocation contains a series of rhetorical techniques- writing techniques that strategically convey the author's purpose and captures the audience's attention in various ways.
Now, audiences are very impatient. Amongst many literary techniques and rhetorical devices, Homer incorporates plot devices to try to structure and organize these anachronistic events. Remember he most likely learned this poem from listening and reciting, not writing it down. The invocation is similar to the Prologue in Shakespeare's "The Tragedy of Romeo and Juliet." Both act as the coming attractions for the audience. This invocation involves one of three important plot devices: foreshadowing, which hints at important later events; it keeps listeners and readers engaged. If audiences are not aware of what is to come and what to expect, they lose focus. By the end of the invocation, audiences know the end of Odysseus's journey, but they have no idea how it happened.