1. The Romans could muster many many legions: the Second Punic War is a Monty Python sketch of this (I raised one army and it was destroyed, I raised another army it and it was ambushed and destroyed, I raised a massive third army and it was enveloped and wiped out, but this army I just raised is the strongest army yet...) Rome commanded massive manpower reserves in both its citizens, but more importantly, in allied states that were compelled to supply armies for Rome and who fought in the Roman manner. So even if you wiped out a Roman legion (or two), another one would be raised the following year to avenge that defeat.

2. It's organization: the organization up until the Late Republic was the Manipular Legion, 3 lines of heavy infantry each subdivided into 10 maniples. In the Late Republic it was reorganized into 10 larger cohorts that were still deployed in 3 lines. The multiple lines meant you could relieve/reinforce the front if needed. The cohort (and maniple to a lesser extent) was a large tactical unit that could maneuver relatively easily on the battlefield. You can see this in Cynoscephalae where an officer pulled 20 maniples to attack the opposite wing of the enemy.

3. The leadership: The Roman general was expected to stay behind his front line and command the army. He was never expected to personally fight - that was a sign that things had gone pear-shaped. This meant he could see parts of the line that were failing and order up reserves or go in person to rally them. There were also 6 tribunes within the legion that had command power - one of them ordered the maneuver at Cynoscephalae mentioned in #2. The legions also had a lot of junior officers, the cohort legion had about 40 per legion (2 centurions and 2 optio's per cohort) for leadership.

4. Experience: Rome was constantly at war and even using a militia system it meant a fair number of the men in any army would have seen prior service. They were still fairly fragile upon mustering (as it was a wholly new formation and needed to develop cohesion) but I'd warrant any civic militia would be as well. By the late Republic the militia element was gone and you had men volunteering for the duration of a war - and with the conquest of Gaul taking a year this meant that some legions were extremely experienced and tough. Under the principate you had a fully permanent standing army that could spend all its time training and patrolling the borders.

Edit: I missed additional factors listed below.

5. Equipment: By the Late Republic the ordinary soldier was equipped to very high standards in comparison to many of their opponents. The Roman state could equip each legionary with a mail shirt - which is a ridiculously expensive piece of equipment formerly only affordable by the very rich. They'd even developed lorica segmentata which required considerable technical skill.

6. Logistics: This is another huge advantage. The Romans in the Second Punic War supplied three field armies simultaneously throughout an entire year - in Spain, Italy, Sicily. Tribal armies had none of that capability and couldn't keep an army in the field for long or even mass a great number as they mostly lived off of the land. Rome could import and move the tons and tons of grain and fodder needed each day over hundreds of miles and across oceans. This gave them an all-year-round capability and the ability to move through territories that other armies couldn't (either because it was desolate or in seasons when the crops weren't ripe).

Not a specialist in this field, but:

1. Robust agricultural system raising plenty of young men as, well, a nearly inexhaustible supply of spear fodder; this especially came on-line after North Africa, although the latter stages of the Latifundia system seriously depopulated the empire with their (probably) deliberate inefficiency (for some of the same reasons that OPEC countries historically under produce.)
2. Military training and collective disciplining system recreating the sons of farmers as soldiers - competent not so much in individual combat but in the far more effective mass formation warfare studiously adopted from the Greeks (and greatly improved with the pilum). Attendant to this the large professionally soldiering class(es), which most other societies around them lacked, such as the Celts - they had the equivalent to the Roman equite, but not the soldiers at large.
3. That formation warfare; fighting as a coordinated body is exponentially more effective than the sort of warfare practiced by most of the Romans' enemies.
4. General emphasis on infrastructure and logistics partially inherited from Roman mercantile practices. They were great traders (read: also army-suppliers), road builders, dock builders, etc. Also their field engineering was second to none - for a very long time.
5. Willingness to learn from the enemy and adopt - and improve - its devices, as we see, for instance, in its naval struggles with Carthage or its tactical nose-bleeding from Pyrrhus. As well as their general innovation and willingness to experiment whether technologically or tactically, especially in times of necessity.
6. Better equipment. Mass manufactured, for the standards of the time, and of high quality owing to better furnaces, etc. Also their relative abundance of imported Wootz steel.
7. Willingness to invade the lands of others. This is a surprisingly underrated "virtue" in terms of military history. Many nations will mark their territory and settle out, discouraging expansionism and effectively programming it out of their military/public mind - because too inconvenient, costly, uncertain, etc. With their other advantages, the Romans could afford to be a warlike people, ready to brave the frontiers for wealth and glory. This motivation is not to be underestimated.
8. Experience. You learn a thing or two when you're at war with, or preparing to go to war with, just about everyone around you for hundreds of years.
9. The reputation which preceded them and the fear it struck. Also their own high morale - not because they were always successful, but because they were so successful on the whole that even the most disastrous failures didn't undo them (though they certainly could have a few times) - what to say of the minor injuries. Their sense of empire and its mission was very strong at its heyday, not unlike, say, Manifest Destiny. Moreover, they were the civilized, the beneficiaries of all real culture, unique and special. Their adversaries were barbarians - dehumanized lessers.
10. Intelligence, in the sense of information gathering. First, the Roman merchants (and later, missionaries) would supply information on prospective conquests or adversaries. Second, because of their logistical bent, the Romans were also healthily obsessed with up to the minute information using messengers and that road system of theirs.
11. Money. Seems almost a non-factor but assuredly was not. Just as England's common law origin of its financial system based on bond trading attracted financiers from across Europe to its banner - and thus the feeding and fielding of its soldiers - Roman financial ingenuity played a large role in both their military and economic expansion.

More specifically, what made the Legion structure work so well?