Rome didn't aggressively expand. It was very unroman to be the aggressor in a war, they only attack in self defense.
It's not Rome's fault that all their neighbors happen to pose a threat to Rome by having the ability to attack, forcing Rome to preemptively attack them in self defense.
The responsibility of governing the territory that was left over after their enemy's defeat just happens to fall on Rome since nobody else was around to do it.