To understand this you need to understand the context religion had in the Ancient world not just as an article of personal faith but as a way to identify with a community, a culture, and a society. State religion and state sponsored cults were to the ancients an equivalent for identity within a state and their sense of patriotism and nationalism.

The Ancient Athenians did not perceive themselves as citizens of a great city, the holders of Greek culture, the founders of Democracy or a traditional naval power. They had to be tricked into building their navy, and at the height of their golden age many of Athens leading citizens were actively conspiring to eliminate democracy in favor of a return to aristocracy. Athenians looked at themselves as the followers of Athena, keepers of her sacred temple. That a citizen tricked them into building their great fleet isn't surprising, since winning the day through cleverness is an expressly Athenian (Goddess of Wisdom) virtue.

At the point Rome and Carthage were coming into conflict the most popular religion in the Mediterranean basin was the Cult of Hercules. This cult spread throughout the trading colonies being established by Phoenician and Greek maritime states. Hercules during life was a traveler and adventurer who bettered himself through great achievements on his journeys. His cult was an aspirational model for those who wanted to better themselves by committing great deeds far from home, so it isn't surprising it took hold in so many trading colonies.

The Roman connection of self to a sense of society and sense of culture were deeply bound to their state religion, the Pantheon of Roman Gods. The city itself was built around the Temple of Jupiter Optimus Maximus, said to be constructed on the very spot Jupiter saved the first generation of settlers by interceding on their behalf from a group of invaders. To an ancient Roman the pantheon, the rites surrounding them and the traditions of the Gods were their red white and blue, their mom and apple pie, their George Washington and the Cherry Tree story. Additionally, the Pantheon, the complex framework of priesthoods, auguries and rites surrounding them were their equivalent to a modern constitutional authority to govern. All acts of state, including elections and declarations of war, were subject to approval by the Gods through these rites and ceremonies (which included the gladiatorial games, often mistaken as a form of pure entertainment).

Looked at in this context the value of the Trojan foundation myth is apparent. The Romans weren't connecting themselves to a group of losers fleeing the scene of their greatest debacle, though that may have been part of their earliest sense of identity because Rome was likely founded by castoffs from other cities. The connection to Troy is a connection to ancient religious traditions, rites, and prophecies. The Roman sense of dignity, fealty, virtue, and authority, in short, the Roman sense of self, is connected to these grand and august traditions passed down from time immemorial. The Romans viewed themselves as the keepers of the flame of these ancient traditions in much the same way modern Americans look at themselves as inheritors of the Western traditions of Greece, Rome and England.

Richard Miles Carthage Must be Destroyed has a very good discussion of the importance of the Cult of Hercules in the ancient Med basin

John Hale's Lords of the Sea: The Epic Story of the Athenian Navy and the Birth of Democracy has some good info on the connection of Athenian Culture to Athena

lan Watson's The State, Law and Religion: Pagan Rome talks about the connection between law and religion in Pagan Rome.

The Aeneid has to be understood not only as a work of literature and epic poetry, but also as a piece of Augustan propaganda which sought to present Roman ancestry in a new light.

From Rome's expansion in the third and second centuries BC, the once small city-state came to include a great number of other cultures including, eventually, the Greek mainland and the remnants of the Hellenistic east. There was a (mythical) history to the eastern Mediterranean that simply did not exist in the West, and the Romans had very little connection to it. However, once their hegemony was established over the region, Rome began to seriously appropriate the art, literature, and culture of the Greeks over the course of the late Republic, though there was no historical claim to any of it. Some members of the Roman aristocracy, like Cato, demonized this influx of eastern culture, and the conflict between east and west became analogous to the conflicts between Pompey and Caesar and then Antony and Octavian.

With Octavian's defeat of Antony and subsequent full incorporation of the eastern Mediterranean into Roman rule, Rome found itself in an interesting position. It must be remembered that the Romans hated kings, and for 500 years had celebrated the end of the Tarquins. Now, however, Rome (and the Mediterranean) had a single ruler - Augustus. Unable to produce legitimacy in Italy, he adopted the iconography and mythos of eastern rulers, largely from Alexander but also to some extent the Persians and Egyptians. This lacked depth, however, as the Romans had no links to the heroic past.

Virgil, in writing the Aeneid, created these links with great detail. He rewrote the character of Aeneas, survivor of the Greek siege (and therefore, ultimately, a winner), as a man who embodied Roman ideals (virtus chief amongst them). He brought his family (and gods) from Troy to Italy and established what would become Rome. This story became intertwined with another story of origin, that of Mars, Rhea Silvia, and Romulus through literature and art. Together, these tales wove a story of Augustus' links not only to Mars, Venus, and Romulus, but to the Homeric hero Aeneas, direct from Greek myth and tradition.

In doing so, Augustan authors (not only Virgil, but also Livy) helped create a new Roman mythology which legitimated rule in the East and West, uniting them into a single narrative.

Paul Zanker's "The Power of Images in the Age of Augustus" (1990) has more on the visual accompaniment in state art for this narrative.

While it's all well and good that we've been discussing the firmly ingrained religious connotations and repercussions, as well as the propagandistic nature of the Aeneid, I think we're missing a key point, which at first may seem very obvious, but warrants in depth discussion, and that is: nation-building. This work is more than a strain of the cult of Aeneas or cult of Hercules; it is more than the legitimisation of Rome's position in Greece; and it is more than Augustus' propaganda as we've discussed it here. As others have mentioned before, all of these are major themes are most definitely involved, but at its heart the Aeneid is a poignant, tragic epic about nation-building.

Let's not forget that Aeneas doesn't get a "happily-ever-after" after his trials and tribulations, like Hercules who achieves apotheosis. This entire epic carries strong tones of tragedy. Aeneas does not get to go home, like Odysseus. His wife is dead. His father is dead. His Carthaginian lover -- whom he did love, make no mistake -- is dead. The very first thing he says when we see him in book 1 is:

"o terque quaterque beati, / quis ante ora patrum Troiae sub moenibus altis / contigit oppetere! o Danaum fortissime gentis / Tydide! mene Iliacis occumbere campis / non potuisse tuaque animam hanc effundere dextra, / saeuus ubi Aeacidae telo iacet Hector, ubi ingens / Sarpedon, ubi tot Simois correpta sub undis / scuta uirum galeasque et fortia corpora uoluit!" (Ver. Aen. 1.94-101).

"Three and four times blessed were those who happened to perish before the faces of their fathers beneath the high walls of Troy! O Diomedes, strongest of the Greeks! Alas that I could not meet my death on the plains of Ilium and give up my life by your right hand, where fierce Hector, slain by the sword of Achilles, lies, where the sprawling Sarpedon, where Simois rolls and sweeps so many shields, helmets and brave bodies of men beneath its waves!"

Those are some heartbreaking opening lines from our hero. Not really the type of words to inspire, for Aeneas is not the inspiring hero of the Homeric epic. He is in no way as lustrous as Achilles or as cunning as Odysseus. Indeed the ideal Roman epic hero -- which Aeneas is -- is coloured with restraint, his virtus tempered in a mien very different from the Greek arete. Time and time again we see Aeneas lamenting his fate and those of his people. But what he does is endure. He is a dutiful man, loyal to a fault to gods, country and family. Where others would waver, he pushes through. Not for himself. Never for himself.

This outstanding trait of Aeneas, this repeated piety, the restrained virtus, is the reason behind Aeneas' eponymity. The loss of his people in the previous war is more a symbol of rebirth than of pessimism, carried on Aeneas' shoulders to a new land and a new life, a journey rife with tragedy but theirs all the same. Even at the end, when Aeneas plants his sword into Turnus' chest, the use of the word condere ("to found" e.g. a city) is explicit. J.W. Hunt says on the subject, "Before the city can be built, the sword is buried in the heart of its opponent: building and burial are curiously fused in the closing scene" (Forms of Glory, 5). This symbol of rebirth, this strange bougonia -- the proliferation of life in the very body of death -- is instrumental to the nation-building focus of the Aeneid.

Later in book 1, Aeneas sees the Tyrians eagerly constructing their city and likened to bees (Ver. Aen. 1.418 - 1.439), juxtaposed just afterwards by the temple in Carthage carved with a frieze of the fall of Troy (Ver. Aen. 1.464-1.493). He is at once in awe and torn with sorrow. Here in the same place is a city such as the one he wishes for his people, as well as the reminder that Troy is rubble. Troy is the bloated corpse and the Carthaginians are the bees brimming with industry.

So too the Roman race, who were destined to rise, to endure, to forge for themselves a new legacy, a new identity, a new nation.

(PS: we're not going to go into the pessimism vs optimism of Vergil's Georgics. I almost went there. But that would be something else altogether, I think)

As a note to the pro-Augustus propaganda present in the Aeneid: Aeneas was thought to be the son of Venus, and Julius Caesar claimed a line of descent through Aeneas. Augustus would probably have wanted the knowledge of the story and the powerful tool it represented forefront in the minds of his people. The entire Julio-Claudian Dynasty gains divine blood if that story is accepted, and with the knowledge that Caesar and Augustus would both be deified, it sets up their whole family as unassailable. To fight against one of their line would always be an act of hubris because you mess with the gods themselves when you do so.