In the history of Rome it has suffered some major losses, but one recurring theme is that the empire has always been able to get back up to recruit another army. How was Rome able to recruit these armies whereas their opponents could not? Where did these men come from, how did Rome pay for them, and why were they so willing to become soldiers when their predecessors had not faired any better? Today Italy is not particularly known for its high population, so did ancient Italy have some kind of advantage over other places? Did Italy have better agriculture to sustain higher populations?
When discussing the Roman Army it's important to make the distinction between the Republican Roman Army and the post-Marian Reforms Roman Army. The Army in the Republican Era was based on a militia system in which landowning citizens, being considered stakeholders of The Republic, were expected to serve in the army during times of conflict. This meant that the entire population of landowning males could be called upon to serve. As such service in the army was seen as a civic duty by Republican citizens. Adrian Goldsworthy in his book "The Complete Roman Army" writes:
"The militia system worked well until the middle of the 2nd century BC. It had two great advantages over rival military systems. The first was manpower. As the Romans had expanded throughout Italy their citizen population had grown. Some communities were eventually granted citizenship, whilst others received Latin or lesser status but still had an obligation to provide troops. Polybius claims that just before the war with Hannibal the Roman Republic and its allies had over 700,000 men registered as of the age with sufficient property to serve in the army. Only a portion of these men was recruited at any one time, but these huge reserves of manpower allowed the Republic to absorb the appalling losses of the Second Punic War and still emerge victorious... The second great advantage of the militia system was the willingness of Roman citizens and allies to submit to the army's discipline and formal command structure."
However, as Rome expanded this system became inefficient. While the militia system worked well for localized conflicts, it becomes cumbersome and impractical for maintaining territorial expansion and dealing with emergencies outside the Italian Peninsula. In 107 BC Consul Gaius Marius enacted his Marian Reforms as a way to rapidly raise forces for a conflict in Africa. Marius eliminated the requirement that soldiers be landowners, and opened military recruitment to that capite censi or "head count". These were non-landowning Roman citizens, who previously would not have had the means to afford their own equipment, nor the privilege of serving in the army. In exchange for 16-20 years of service (depending on the era) Roman soldiers would have their equipment provided for them as well their own farmland upon successful completion of service. The results of this were two-fold. The first result was the creation of a standing army, made up of long serving citizens who could leverage their years of combat experience to increase the military effectiveness of the Roman legions. The second was to make the army an attractive option and a means of social advancement for poorer Roman citizens. As the ranks swelled with professional soldiers Rome was able to field armies further from Rome itself, as well as maintain garrisons in conquered territories. While Rome continued to expand, and citizenship was granted to more people, service in the army remained a way for Roman citizens to maintain a steady career while having the possibility of joining the landowning classes upon successful completion of their term of service.
The theory I've read (by Polish historian Adam Ziółkowski in his Historia Rzymu - History of Rome) explains it in the following way:
Starting with the conquest and destruction of Veii (396 BC) Rome started to develop a unique socio-political system, where virtually every citizen was a landholder. Therefore, all citizens shared the burden of military service, but also participated in division of spoils.
That was huge. At the time, leading powers of the Mediterranean - the Hellenic monarchies and the Carthaginian Republic - relied on mercenaries and professional soldiers. Unlike Romans, the vast majority of their subjects/citizens didn't directly participate in the war effort and didn't have a personal stake in it.
As a result, both the upper and the lower classes of Rome become united in support of aggressive, offensive warfare. Continuing successes created a snowball effect: each new war brought new spoils and new lands, allowing for the expansion of Roman recruiting base and encouraging future aggressiveness.
As Rome grew bigger, the system began to change. The "big" wars (like the conflict with Pyrhhus) benefited the upper classes, but not the average citizens who didn't want to move out of the immediate vicinity of the city. To solve this problem, a new strategy was developed, where each "big" war was followed by a small conflict with a previously allied or neutral community close to Rome.
Roman historiography calls these small wars "rebellions" and they're usually glossed over by historians. However, Ziółkowski points out that the supposed rebels waited with their insurgency until Romans were free to deal with them, were hopelessly outmatched and - sometimes - not even subjects or allies of Rome at the time!. Therefore he considers these conflicts to be deliberate Roman provocations started for the benefit of the lower classes.
At some point, the ideology of Roman imperialism began to develop. All aggressive actions of Rome were presented as defensive in nature, and past successes were considered to be an evidence of divine favor and fundamental rightfulness of Roman cause. This explains famous Roman tenacity: if their cause was just (and they knew it was), then they would eventually prevail despite initial setbacks. See: Samnite Wars, War with Pyrhhus, Punic Wars.
An additional question: pre-Marian reforms how was the military population wealthy enough to be able to afford their own equipment and weapons?
I think a large difference between Rome and say Athens and Sparta was how citizenship worked.
In Athens, Pericles passed a decree in 450 that to be a citizen both of your parents also had to be citizens. It is thought (in City of Socrates by Jon Roberts) that this decree was passed to prevent poor citizens from marrying their daughters to wealthy metics. A metic was someone who lived in Athens who wasn't a slave but wasn't a citizen either. They were foreign born men and most of them were traders (working a trade was seen as undignified for citizens). Athens, being a democracy, didn't want to expand citizenship too greatly because all citizens had a say in how the city ran. The infantry in the Athenian Army (hoplites) were always citiztens. They couldn't call up another army if the first one lost.
Sparta's population was affected by their brutal system. Infanticide was institutionalised, women married late and soldiers were expected to stay and die. All this meant that by 330 Aristotle believed there were only 1,000 Spartans left.
Rome was different to Greek city states in that citizenship was not as closely guarded. If you lived in Rome and you were not a slave or a former slave, then you were a citizen of Rome. If you were the son/daughter of a former slave then you too gained citizenship. In 212 The Edict of Caracalla declared that everyone in the empire who was not a slave was a citizen.
StupidGuns mentioned the Marian reforms which are very important. In 122 Gaius Gracchus was elected Tribune of the Plebs in Rome. He proposed highly ambitious legislation. He provided subsidised grain for poor Romans and opened up provinces for wealthy Romans to take tax payments. Most controverisally, he proposed expanding the citizenry to include the rest of the people living in Italy. Gracchus was eventually killed because this measure was unpopular with the people of Rome who didn't want to share their grain subsidy. These disputes eventually led to the rest of Italy revolting in 90 BC. In order to put down the revolt the rest of the Italians were offered Roman citizenship. This meant Rome had a way larger force than any other state in the Mediterranean. This meant Rome could continue taking over other smaller states and they could then absorb their culture.
Rome also adopted the practises of other people. For instance, the swords they used were called "Gladius Hispaniensis" or Spanish sword.
TL;DR: Rome was better at adapting and more open to innovation than the Greek city states. This allowed them to create a Mediterranean empire.
Part of the answer lies in the sheer size of the Roman population. Once the Roman empire had acquired enough land and people, opposing nations simply would not have been able to compete.
Since Rome wasn't at war on every border at the same time, it could bring it's very large, well trained and well funded armies to bear against whoever it was fighting. And if something went wrong, Rome's population and coffers were deep enough to absorb the hit.