In answering this, we need to be aware that Greek and Roman myths are interrelated in two basic ways. There is the obvious relationship of the Romans having consciously borrowed Greek mythic elements and transferred them to their own beliefs. But on a higher level, Greek and Roman myths can be viewed as having a common origin, just as Indo-European languages do. A kind of proto-mythology possibly existed from which the later individual myth systems evolved (Greek, Roman, and even Norse), analogous to the hypothetical proto-Indo-European ancestor of the modern languages.
In The Aeneid, we can see both these relationships overlaid by Virgil's interpretation of them as connected with the history of the Roman people. By the first century BCE, the Romans had developed a consciousness of their own mission as one of leading and unifying the "world." It made sense to them that the gods had endowed them with this mission in the form of a mysterious prophecy, which Virgil alludes to. At the same time, Romans self-consciously absorbed the intellectual basis of Greek culture just as the Republic militarily and politically absorbed the former Greek city-states into the incipient Empire, which became a reality during the period Virgil wrote The Aeneid.
This was a kind of paradox. The Romans had conquered the Greeks but ironically could not have had their own achievements without Greek culture as their foundation. The Aeneid's starting point is the primal mythic and historical event of the Trojan War. The Romans saw history as a circular process in which a hero, Aeneas, whose people have been defeated by the Greeks, is then destined to found a new people who in turn will defeat the Greeks. The process is a kind of metaphor of the ultimate unity of the Mediterranean world. In it is contained, so to speak, the common basis of the Greek and Roman myths, their specific manifestations as recorded in literature, and the reality of what would later be called the Greco-Roman world.
Virgil takes the existing mythic-historical sense the Romans already had and anchors it in the specifics of his own era. It is not just the intention of the gods that Rome will lead the world and bring peace to it, closing the "iron gates of war," but that Julius Caesar (the great-uncle of Virgil's patron and friend Emperor Augustus) will be the focal point and driving force of this goal. Western civilization as a whole has implicitly adopted Virgil's interpretation of history, given that the beginnings of Christianity, ironically perhaps, occurred only a few years after Virgil wrote his epic.