To ask how the Roman gained the support of conquered people is almost to answer the question. To be sure, Rome had a fairly steady hands off policy, which was probably a great move. For example, they allowed for the freedom of religion and even self-rule, but if people got out of line, Rome's military would intervene. So, I think the best answer is people had to support Rome, because Rome conquered them and they would do so again if these people got out of line. That said, the Romans also had incredible political skills. For example, when they were in Greece, they created a slogan - "Freedom of the Greeks." Titus Quinctius Flamininus was the consul there and through this policy of freedom, he gained the love of the Greeks. However, from a broader perspective, it seems as though Rome was imperalisitc. In the end, I think the best answer is found in the question. They won people over by winning. Also human nature says that if you cannot beat someone, you might as well join them.
The main thing that the Romans did to get the support of the people they conquered was to treat them (especially the elites) relatively well.
When Rome conquered a territory, it was, for the most part, run by its own people rather than foreigners from Rome. Rome did not try to abolish their local religions or kill the local elites. People in the conquered territories could, eventually, become citizens, especially if they served in the Roman Army.
So Rome treated the people pretty well and in return, they generally supported Roman rule.