Rome’s most potent weapon appeared to be ‘enfranchisement’. They brought upon Italy in particular, but also their other, later territories the rights and privileges of Roman citizenship, or at least good government, security and a reasonable justice system.
In short, Rome was a state, not a tyranny. In fact during most of its expansion, Rome was in fact a republic, not governed by an emperor, but by the Roman senate.
People preferred to be ruled by Rome for the alternatives were either wild barbarians to the north or tyrannies to the east and south.
Also some kings without heirs left their kingdoms to the Romans, seeing Rome as the best governor of their people after their own death. Kingdoms like Cyrene and Pergamum fell into the Roman possession by such peaceful means.
Also, empires of that time broke apart if they were attacked by a foe. The oppressed peoples under the yoke of the dominant power rose up and joined the foe. This happened when Rome attacked others, but failed to occur when others attacked Roman territory.
Naturally, the famous Roman army is also a major contributor to the building of the empire. In rough sequence, first they defeated the Etruscans, then the Samnites, then the Greeks (or better a Greek invader called Pyrrhus), the Gauls in northern Italy, then the Carthaginians, then the Macedons, then the Syrians…
Particularly by defeating the latter three, they defeated the remaining significant powers who could have been empire builders instead of them. It is much debated how they actually got into all these wars. Generally, it is understood that they were brought into all these conflicts by either themselves or their allies being attacked. And by winning they so acquired the vanquished foe’s territory.