I know that they had almost impeccable morale, meaning that they rarely fled, but was it something in their training that made them so strong against other armies?
Several reasons:
As /u/rrl pointed out, Rome possessed massive reserves of manpower. In the middle Republic, half of any field army was composed of Italian allies who were mostly organized and fought in the Roman manner. This meant even disasters like Cannae, only half of those were Roman casualties. More importantly Rome did a great job of incorporating those allies and getting them to 'buy in'. Despite handing Rome three consecutive losses, Hannibal only managed to sway a few Italian states over to his side. In the late Republic the Italian allies would revolt, not against Rome, but to gain Roman citizenship. This meant every Roman conquest added to its pool of manpower.
as /u/lapzkauz noted, Roman logistics was top notch. This can't be emphasized enough. In a world without rail or trucks, supplies had to be shipped, or carried/pulled by men or animals. Obviously the former is most efficient but only for areas by the coast. Once you move inland, you need to carry food and supplies. Most states/nations/peoples lacked the sophistication and relied on looting the surrounding territories for food, when that dried up, their army had to retreat. During the Peloponnesian War, the Spartans regularly raided Athens but could only stay for a few weeks. The Romans however could move hundreds of tons of supplies deep inland throughout the year (drawing on their entire empire). Roth estimates each soldier required 2.2kg of food each day, so roughly 10t of food needs to be transported to each legion (not counting their horses or pack animals) on mules, donkeys, or oxcarts - that themselves need to be fed each day as well. This meant they could go through regions other armies couldn't, move at times when others couldn't (before the harvest), and keep armies in the field year round.
Command and control. A historian noted that the classical Greek phalanx was an army with the lowest rank being Captain. While having their generals fight in the front rank was good for morale, it meant he couldn't control his army once the fighting started. Roman generals were not expected to fight (if they had to, it meant things were going truly pear-shaped) but to stay at the back controlling the army and noting acts of courage (or cowardice). This meant they could react during a battle, send in reserves to hard-pressed units, rally units.
Discipline. The Romans were ruthless in this aspect even coining a word specifically (decimate). Whether the story in Livy about the consul who executed his own son for disobedience ever happened or not is unimportant - that was what the Romans believed had happened.
Tactical system. Pilum plus sword and shield, the manipular and later cohort units gave the Romans tremendous flexibility and maneuverability over the previously dominant sarissa phalanx.
Siege warfare. No one could match the Romans in their day at this. Consider how Caesar built up a 12mi long wall to invest Alesia and then threw up a 13mi outer wall to defend against a Gallic relief force. But the ultimate definition would be Masada - Herod had built a fortress palace at the top of a rocky plateau in a desert and the Romans looked at that and said "we'll take it".
Refusal to accept anything except victory under their terms. With Hellenistic leaders, you might lose a big battle and be forced to surrender. Give up some concessions and plan for revenge in a generation when you've rebuilt your army. Rome just never stopped fighting until the opponent said 'uncle'. The 2nd Punic War pretty much exemplifies this: Rome had lost 3 consecutive armies (including a double sized one), and with Hannibal marching on Rome, were scrapping together a new force. At this stage most states would have started negotiations, maybe give up Spain and maybe even Sicily. Rome just said 'nope' and continued fighting. They even exiled survivors from the Cannae army to Sicily for the crime of not winning - at the time Hannibal was still romping up and down Italy. Rome would just make the other guy blink, despite being stabbed 5-6 times, shot, run-over, and poisoned.
Lastly, Rome had some really good leaders. There is a tendency to downplay Roman elite generals and ascribe the army's success to the Centurions which we equate with sergeants - which is incorrect (they were mostly appointed from the equestrian class). While the Romans had a lot of stinkers, every army in every nation did. For every Patton, you'd get a Fredendall. Rome however had some exceptional commanders: Caesar, Scipio Africanus, Septimius Severus, Trajan, etc...