In what ways does the Aeneid fulfill its aim to provide the Romans with a national epic?
Virgil's Aeneid does provide the Romans with a national epic because it establishes a link between their current history and their mythical past. As early as Homer's Iliad 20, the story existed that Aeneas' descendants were destined for greatness. Virgil expands upon the hints given by Homer by showing the link between the divine (Venus), the mortal (Aeneas, Ascanius/Iulus), and Rome. Aeneas' son Iulus would be claimed by the Roman Julian clan as one of their ancestors, thus linking them and Rome with Venus. Thus, people like Julius (spelled Iulius in Latin) Caesar and his adopted son Octavian (later known as Augustus) could claim to be descended from the gods:
From this glorious source a Trojan Caesar will be born,
who will bound the empire with Ocean, his fame with the stars,
Augustus, a Julius, his name descended from the great Iulus. (A.S. Kline translation)
Augustus, of course, ruled Rome at the time when Virgil was composing the Aeneid. On Aeneas' shield in Book 8, Virgil even depicts Augustus triumphing over Antony and Cleopatra at the Battle of Actium.
Aeneas' encounter with Dido in the first half of the epic, also creates mythical links with the Romans' historic past. Virgil's Roman audience will have known that a relationship between Dido of Carthage, a city which the Romans destroyed in 146 BCE, and Aeneas, one of the mythic founders of Rome, would never be successful.
Virgil's audience could well believe that Jupiter's prophecy in Aeneid 1 had come true, because they could see the results with their own eyes:
I’ve fixed no limits or duration to their possessions:
I’ve given them empire without end. (A.S. Kline translation)
Thus, with Aeneas' triumph over Turnus at the conclusion of the Aeneid, Aeneas was well on his way to fulfilling the prophecies set forth in the opening of the epic.