The Roman origin myth included a descent from the Trojans via Aeneas, not from Greeks, so surely that wasn't how they explained it. How then, seeing that they were highly infatuated with Greek culture and had much familiarity with the Greek language and must have noticed many similar words in Greek and Latin, did they explain that fact? Conversely, how did the Greeks explain it?
A comment in a previous thread linked this paper, which says that some philosophers in the Roman-era did try to explain the similarity between Greek and Latin as Latin being a divergent Greek dialect.
I can get this started, but someone who knows more will have to continue. The short of it is that there was a lot of cultural diffusion and selective borrowing by the Romans from the Greeks. Keep in mind that while the Greek world was at its apex, what is now Italy was still Etruscan. Also, Greeks extensively colonized southern Italy and Sicily throughout the Hellenic period so it makes sense that they would bring their religion and way of life with them.
During the days of the Roman Empire, though, it was a bit different. Greek culture and language was given a premium by educated Romans and Greek tutors were very much in demand by wealthy Romans - similar to how Latin was viewed in Europe in the Middle Ages. Because Greek culture had been the dominant culture of the Mediterranean world for hundreds of years before the rise of the Roman Empire, Greek was actually considered the lingua franca of the time, and not Latin. Thus, if you were a worldly, educated Roman, you had to know Greek. It's worth noting that all of Paul's letters from the New Testament were originally written in Greek - exquisite Greek.
Tl-Dr: Greek had been the lingua franca of the Mediterranean world for hundreds of years already before the rise of Rome, so Romans were keeping with the times.