The myth of Aeneas founding Rome, was mainly popularised with the publishing of the epic poem "The Aeneid", written by Virgil. This was written between 29BC and 19BC. Aeneas was already a myth to the Roman's, having been mentioned in the Illiad, however virgil solidified the character, his vague association with Rome and made him into a national epic. The Aeneid is the story of Aeneas, a demigod, son of Venus who escapes from the siege of Troy, and goes on various adventures around the Mediterranean, most famously encountering Dido in Carthage.

An important thing to note, is that since Virgils birth, he had constantly grown up surrounded by war, after the death of julius Caesar, to war with Mark Antony, culminating in Actium. Octavian, (later Augustus), would close the gates of war in Rome, which had been open for multiple decades, symbolically bringing peace across Rome.

Virgil had already written several famous poems, and he would have been known across Rome (mainly for his collection of poems the "Ecologues"). It is believed, he was commissioned by Augustus, to write a new story, symbolically re explaining Romes rise, in order to legitimise his rule.

Throughout the book, Aeneas grapples with his "fate", Gods frequently appear to Aeneas, imploring him to make way to Italy (this was straight after the Trojan war, no concept of Rome existed at this time). Aeneas struggles with his belief in the gods "Pietas" against his natural anger "furor".

This in itself can be seen as a praise or criticism of Augustus' regime, which is a whole other issue that dosent need explaining for your question.

Anyway, Aeneas throughout the poem, encounters references to his descendants, through armour the gods give him and what the gods say to him. Then he travels to the underworld, where he is shown that his line will bring. He is shown Romulus and Remus, the 7 Kings, the republic, Julius Caesar and finally, it culminates in Augustus. Who is described to be shining, and to represent a new beginning for Rome, just as Aeneas is THE beginning for Rome.

Eventually Aeneas travels to the birth place of Rome, then known as Latium, he has a war with the rulers there for the daughters hand in marriage, so he can one day rule. This is a massive oversimplification, but Aeneas kills his chief rival, and that is when the book ends.

I believe this abrupt ending is intentional, however some would disagree and say it was due to Virgil not finishing the book (Augustus had it published post-homously).

Regardless, the reader can infer that one of Aeneas descendants (from the underworld chapter) would be Romulus and Remus, who would found the physical city of Rome. This was all done so Augustus could trace his lineage back all the way to the Gods and time of Ancient Rome and Troy. This legitimized his rule in the eyes of many.

There are a lot of other themes the Aeneid, however this is the principal one relating to your question. There is some debate, as to whether the character of Aeneas was written as a true hero, but was actually a criticism of Augustus himself.

In essence, Aeneas traveled from Troy, to the ancient site of Latium, eventually his descendants, Romulus and Remus, would create the physical city. But the implication is the spirit of Rome was born with the start of Aeneas' bloodline.

Sources: Virgil. West, David. (1990) "The Aeneid". Penguin books. London. Heinze, Richard (1993). Virgils epic technique. University of California Press. Berkley. Various lectures I have attended over the years, and my notes from them. Edith Hall, and Boris Johnson.