What is the purpose of the invocation at the beginning of the Odyssey?
The Odyssey by Homer, as is the case with other early Greek epics, including The Iliad and Hesiod's Works and Days and Theogony opens with an invocation of the Muses.
The Muses, daughters of Zeus and Mnemosyne, the goddess of memory, were the goddesses who were patrons of the arts, including history, dance, and poetry. Calliope, in particular, was the Muse associated with epic poetry.
The first aim of the performer in invoking the Muses was to gain their favor and ensure a good performance. Just as a farmer might offer a sacrifice to Demeter or a mariner to Poseidon, for a bard to honor the Muses was practical, asking the favor of the appropriate divine patron.
The second purpose of the invocation was to convey to the audience that the song the rhapsode would perform was not a unique, idiosyncratic composition, but rather a traditional tale, handed down over generations, gaining its authority from the divine Muses who inspired it. Rather than the poet claiming individual inspiration and credit, as a modern poet would, the ancient bards are emphasizing their role as a conduit of the divine; the Muses speak through the bard.