Romans imported their silk from China, mostly with Persian merchant as the middlemen. As a matter of fact, the Silk Road ended in Rome in the West.

The Romans didn't know that silk is actually of animal origin, they were convinced that it was obtained from trees. Production of silk in the Mediterranean began much later in the 7th century when the Eastern Roman Empire went as far as to send spies to steal the silkworm eggs, which were a closely guarded secret (as described by Procopius).

So, just adding to the other comment, the "Silk Road" wasn't exactly that much of a thing, and the Romans would get silk through Parthians, Arabs, Scythians and sometimes even Indians. Silk came not as a fabric but as a substance, as is noted by Pliny when he writes "[the Chinese burden] our women with the double task of unraveling the threads and weaving them together again". Silk was famously from China, or as the Romans called them the "Seres", which translates literally to "silk people". Pliny also claims that silk is, as /u/EmperorOfMeow wrote, that silk came from trees that the Chinese would soak then rake the substance off of, so Romans really knew quite little about silk for quite some time. Greek and Arab writers/traders on the other hand tended to know that silk came from worms, as did later Roman writers, but AFAIK they didn't (successfully) bring silk production to the West.