the Romans weren't bent on wiping out other cultures. They typically left people their own religions, for instance, merging Roman gods with local ones like Sulis Minerva at Bath. They wanted land and they wanted trade. Other people wanted to trade with them too.
So part of it is economic success. Part of it is because when they did go to war they were better at it. Part of it was engineering; they didn't invent roads (there were even roads in Britain before they arrived) but they built good ones that allowed traders and troops to get around efficiently.
But in the end imperial expansion is a Ponzi scheme; you need ever increasing land to feed ever increasing growth and something has to give. But they lasted 1000 years in the west and another 1000 in the east, which is astonishingly successful, and until civil wars became endemic they administered it all pretty well.
Re barbarians, it's a Greek word meaning anyone who didn't speak Greek, all they could say (it seemed to untutored Greek ears) was baabaabaa. It also became a word meaning bearded, as many barbarians had them - so it's related to the word barber. But many of their enemies were barbaric. They were horrified by the British priests practising human sacrifice.