Both the Ancient Greeks and Romans were descended from Proto-Indo-Europeans. While the two groups had diverged, they continued to share remnants of a common language and other features including mythology.
The most obvious sign of this is the chief deities of their respective pantheons: Zeus and Jupiter: both derive from the Proto-Indo-European sky deity *Dyēus ph2ter. Their counterparts could be found in various Indo-European mythologies, including the Germanic Týr, the Baltic Dievas, and the Vedic Dyaus Pita.
Thus, Greek and Roman mythologies exhibit great similarities due to them being cousins.
Furthermore, from the earliest period of the Republic, Roman religious belief had adopted Greek elements. This begun extremely early, and far predates the Roman conquest of Greece. One example is Apollo, who was directly adopted into the Roman pantheon. A temple for him was erected in Rome as early as 431 BC, long before the Romans conquered Greece in 141 BC.
One way in which Greek beliefs were transmitted to the early Romans was via the Etruscans. After their conquest by the Roman Republic, elements from the Etruscan belief system were integrated into classical Roman culture. In this way, the Etruscans bequeathed notable Greek influence on their own mythology to the Romans.
For instance, the Etruscan Menrva passed on influences from the Greek Athena to the Roman Minerva. Another example is Mercury, who inherited traits of the Greek Hermes from the Etruscan Turms. Both long predates the Roman mastery of Greece; Mercury had a temple on Aventine as early as 495 BC.
It is not accurate to say that Rome inherited Greek religions because she had "little mythology" when she conquered Greece, although Greek culture did went on to be highly influential on Rome ("Captive Greece captured her rude conqueror").