How did the seasons change, according to the Romans?
The Romans believed the same thing, actually.
In the Roman version, Ceres is the goddess of agriculture and mother to Proserpina, an obvious copy of the Greek Persephone. Like her Greek version, Proserpina was abducted by Pluto, and Ceres caused the earth to stop growing food until her daughter was returned at the behest of Jupiter, but not before eating seeds from the underworld.
Most prominent [of Italian rites connected to fertility] were the annual rites to Ceres, which were performed at all crossroads in the city by Roman matronae, commemorating the kidnap of Proserpina, and Ceres' search for her daughter which resulted in the story explaining the changing seasons.
- Lennon, Jack J. Pollution and Religion in Ancient Rome. Cambridge University Press, 2013.
Rome imported Persephone's story from the Greek colonies in southern Italy around 205 BC.