How did the Greek gods become Roman?

Answer 1

During the Bronze Age, The Greeks were very busy colonizing the Eastern Mediterranean. After the fall of Troy (itself a Greek colony), the survivors fled across the sea to the west, eventually settling on the Italian Peninsula. However, there were already people living there when the Trojan refugees arrived, namely the Sabines, Etruscans and Latins. Over time, because of the common source language, and the fact that myths represent natural phenomena anyway, it didn't take long for their religions to merge.

That said, there are plenty of myths and gods that are purely Roman, like the story of Cupid and Psyche or the god Janus.

Answer 2

They didn't. The Greeks had their gods and the Romans had their gods.

The Greek and Roman gods are almost the same except for their names and their personalities.