The Aeneas myth predates Augustus by a long way. It's in Cato, the earliest Latin historian. It's in Ennius' Annals, Rome's first great national epic, now mostly lost unfortunately. Basically it's as old as Roman literature.

Does it have anything to do with fact? Well, since most of ancient scholars today think at best the Trojan War stories can't be shown to reflect what really happened, and most likely don't reflect what really happened, it follows that the Aeneas myth can't either. How can you have a real refugee from, say, the War of the Ring?

That's not to say the Aeneas myth was a silly fiction. Cato and Ennius both lived through Punic 2 and Rome's first advances into Greece. How Rome related to Greece was a hot issue. You find Cato, for instance, in some ways rejecting Greek culture. He is supposed to have written his history in Latin because he didn't want his son learning Roman history from Greeks. In other ways Cato seems to have adopted a fair bit of Greek culture: writing history before Cato was a Greek activity, and in fact other Romans who had written histories like Fabius Pictor wrote in Greek. Cato famously refused to speak Greek on his visit to Greece, but that of course implies he could speak Greek.

Romans were interacting with Greeks in other ways too. The earliest Roman poet we know, Livius Andronicus, adapted the Odyssey into Latin. The Latin comedians seem to have stolen plots and themes from Menander's New Comedy.

The Aeneas myth was a way for Romans and Greeks to contextualize this complicated relationship of culture and power. Romans had a place in the Greek past; they fit in the story. This is a point explicitly made by Dionysius of Halicarnassus in the first century BCE. Dionysius, living in a world where Rome had just wiped out the last Greek kingdom (Egypt, at Actium in 31 BCE), was trying to square the military domination of Rome with the perceived cultural superiority of Greeks. Dionysius argued that Romans were a confluence of Greeks who migrated to Italy before the Trojan War (at first called Aborigines) and the Trojan refugees. Dionysius traced the Trojans back to Greece through Dardanus, who lived in Arcadia, a district of Greece, before moving to Anatolia. So Romans were really Greeks on both sides! So Rome's military superiority and Greece's cultural superiority presents no contradiction.

From time to time we do hear about an Anatolian connection to Rome outside the Aeneas myth that is sometimes used as a confirmation of it; that is the Etruscans. The Etruscan language is baffling; it's not related to the Italic languages. Sometimes we see articles arguing it's Anatolian in origin such as Adrados, "Etruscan as an IE Anatolian (but not Hittite) Language" JIES 17, 1989, pp. 363-383. I'm not a linguist; I don't know how well founded such theories are. I think it's fair to say they haven't penetrated Classics yet; Etruscans for us are still a mystery. Likewise from time to time we get genetic studies that claim an Anatolian origin for Etruscans, such as this article in the Guardian. I'm not a geneticist, so I can't comment on the science behind these sorts of articles, but I have never run across an article in an academic context making this sort of claim.

EDIT: fixed a stupid brain fart.