The first issue here is one of periodization. Although Homer's epics were composed in ancient Greece, they were not composed in the "classical" period which is usually described as running from approximately 510 BC to 323 BC. Instead, the invocation to the Muses was a feature of pre-Classical Greek poetry from the period between the fall of the Mycenaean empire and the beginning of classical era.
After writing was lost with the collapse of Mycenaean civilization, the dominant literary mode became oral epic, a genre improvised in live performance. In this setting, the authority of the bard or rhapsode depends on ability to transmit important cultural truths accurately.
The performer invokes the Muses first because he wants their help, as they are the goddesses in charge of poetry and performance. Next, the invocation impresses upon the audience that the performer isn't just speaking as an individual but is divinely inspired and transmitting special knowledge beyond that accessible to mere mortals; the Muses are described as speaking through the rhapsode. Finally, as freelancers paid by rich nobles, emphasizing that they were favored by the gods was probably a good indirect strategy for rhapsodes to hint that it would be a good idea to tip generously.