Good answers in these threads should help

Rise - https://www.reddit.com/r/AskHistorians/comments/1civyn/what_were_the_reasons_for_the_rise_of_rome/

The Army - https://www.reddit.com/r/AskHistorians/comments/3c3zy6/why_did_the_romans_have_such_a_numerical/

FAQ - https://www.reddit.com/r/AskHistorians/wiki/antiquity#wiki_the_roman_empire

Basically, Rome was succesful at "assimilating" different city-states into the Roman state, which gave it a huge manpower pool (700k men at the 2nd Punic War). The subjugated Italian states, and those who had joined Rome willingly, the Socii Italici, paid taxes and gave soldiers for Rome. Another factor which affected the manpower was the fact that unlike, for example, the Diadochi (Alexander the Great's hellenistic successor states) like Ptolemaics and Seleucids, the Romans levied troops from almost the whole population (excluding the capite censi who could not afford equipment). The Diadochi, on the other hand, had to rely on the rather limited amount of Greek and Macedonian colonists in their kingdoms for men to stand in their phalanxes. The amount of manpower Rome had gave it the possibility to shrug off most of the disasters she faced, like the Battle of Cannae, to name a famous one. Losing 60k (smallest statistic given on the wikipedia article) would have put any hellenistic state on it's knees.

Another factor was that Rome was very good at adopting techniques, tactics and technologies they came across. They constantly adapted to new circumstances. For instance, Romans had no navy to speak of before the 1st Punic War.

In addition, the lands of Italia were very fertile and populous, which also added into the Roman manpower pool.

I wrote most of this from what I remember from Adrian Goldsworth's The Complete Roman Army, I'll look through this post a bit later and give more exact info.