This is a great question. As you state, Rome did not start off as a great power. Rome start off small and far less powerful than their neighbors. However, as you also know, Rome ended up a superpower. What happened?
First, Rome was surprisingly well at incorporating other people groups. For example, in Livy, the Roman historian, we read of the rape of the Sabine women. It is a tale that shows that the Sabines were incorporated with the Romans to a certain extent. In short, the Romans were able to absorb other groups and for this reason became more powerful.
Second, the Romans also possessed great military abilities as well as good statesmanship. According to Polybius, it was their constitution that allowed them to form into a great people.
Third, this is an important point that is not emphasized enough. The Romans believed that the gods appointed them to rule. We might call this a self-fulfilling prophecy, but in any case, the Romans were dead serious that this was their destiny.