Also, remember that there were Greek colonies in Sicily, Corsica, Sardinia, and perhaps in mainland Italy as well. So there were surely Italians already worshiping those Greek gods before the Romans conquered the whole area. So they didn’t have to import the religion all the way from Greece, there would have been local priests and sybils practicing it in their own lands.
The Greeks took their Pantheon every where but Romans did travel to Greece to learn from them. And they did lift the Greek Pantheon almost wholesale. A few minor changes where we can almost see the original Roman gods peeking out from behind the shrubbery and some rather interesting naked patches where the fit is not quite perfect but they took a lot from the Greeks.
And why not? Greece was the great civilization. One of the lessons they learned was "Write it down" which is where the Twelve Tables of Law came from. Actually they learned what NOT to do in this case because Greece tended to base their laws on custom rather then an explicit contract.
And all of that eventually led to the Constitution we have today.