While being a Classicist, Latin literature is not my specialisation, but I'll try to answer as best as I can and maybe someone a bit more knowledgable about Vergil can chime in.

When Augustus became emperor he did his best to make the rule of a single emperor palatable for Roman citizens. One way of doing this was referring to the old customs that made Rome great in the past and he was very keen to place himself in a somewhat continous history that started with the foundation of Rome and lead to his reign. Augustus therefore not only initiated an avid building policy (the ara pacis, a temple built to celebrate Augustus as bringer of eternal peace and prosperity, for instance, also includes one panel that shows Aeneas) but also fostered poets to praise him and his reign. Vergil was one of them and the Aeneis was written to, in some extend, praise Augustus. The epos aims to establish an interpretation of history where the deeds of Aeneas and all of Roman history lead to one specific point: the reign of Augustus.

At that time, the myth of the Aeneas was therefore pretty well-known within the Roman public sphere and has been before. The figure of Aeneas was connected with Italy and Rome since the 5th century BCE and at around the First Punic War a few aristocratic families started to trace their ancestry back to Aeneas, especially the Julian gens (the family of Julius Caesar, who in turn was the adpotive father of Augustus). Authors before Vergil such as Cato and Naevius wrote about the myth before Vergil took it up.

As to the motives of Homer and the Odyssey, Aeneas is only briefly mentioned in the Iliad, and while things like composition and structure are influenced by Homer, the plot is not based of Homer. Dido for instance is not mentioned in the Odyssey, it even seems that initially there was no conncetion between the myth of Dido and the of Aeneas.