In a sense, there were many reasons as to why it was Rome that grew:
1) Geographic location: Rome's geographic location allowed there to have natural barriers, yet have areas that allow Rome to expand. Key areas were defended (or was supposed to be defended) well, and Roman infrastructure and advanced-technology building allowed speedy intervention when needed, whether during offensive season or defending against invasions
2) Trade: Rome's trading capital was large, as they were able to incorporate large amounts of different people together into one system, and help each of them excel as well as grow their economies, by enlarging their trade sources, and building up cities.
3) Technological Advancements: Warfare was extremely important to Rome, and Rome was able to produce advanced weapons, as well as upgrade older technologies, ranging from the pilum, catapult, corvus (which allowed the Roman ships to hang onto and board enemy ships.
4) War Tactics: Roman war tactic allowed their empire to stretch from Briton (Present day United Kingdom) all the way to the Asias. This usually incorporated troops into testudo's, which was an interlocking of shields to block enemy arrows, incorporation of Roman troops and calvary to defeat enemies at decisive moments, as well as projectile weapons.
There were many other reasons that are not listed in this list.
~Rise Above the Ordinary