In the centuries after the death of Alexander, the successor kingdoms that sprang up following his death were still great powers in terms of their wealth, resources and military strength. Macedon, the Seleucid Empire and Ptolemeic Egypt were incredibly powerful nations and their armies seem to match Rome in terms of numbers, diverse troops and generals capable of leading them (Hannibal, the famous Carthaginian general, even led Seleucid forces for a time during his exile).
So why were these former great kingdoms no match for Rome? Egypt was the breadbasket of the Mediterranean, the Seleucids occupied a strategic place on the Silk Road and Macedon was still the predominant power in the Aegean and exerted great influence over the Greek city states.
I am aware that at the time of their conquest, all three powers had gone through a period of great decline and in particular, the Seleucids and Egypt had been wracked with political instability. But the question remains, how was Rome able to overcome all three great former empires?
There are a couple of possibilities and the importance of various factors is up for debate but I think two are particularly important:
First, the Hellenistic kingdoms relied on a balance of power. You have three major players and a handful of small players and they are constantly rearranging their alliances and vying for hegemony but at the end of the day they realized that balance of power is best. Nonetheless they keep vying for hegemony because military action and success is demanded as a result of their political systems. This is the second reason. Their kings had large mercenary armies that needed to be kept busy and well-payed and the kings themselves needed to maintain legitimacy and campaigning was a good way to do that.
Rome breaks that balance of power by being a fourth player. They start by propping up Egypt and their approach in the Hellenistic east is to maintain the balance of power and cultivate client kings. They are very successful but they do have constant revolting and eventually they do away with the monarchs altogether ending with Egypt in 32 BCE.
Rome was able to compete for two main reasons: the first was the acquisition of Spain after the Second Punic War (they also proved able to beat Hellenistic armies in the Pyhrric Wars in 270s and in the First Macedonian War which took place during the Second Punic War). Spain gave Rome all the mineral wealth it knew what to do with and the security of the Western Mediterranean allowed the population and wealth of Rome to soar giving them huge manpower and a deep appreciation for the windfall conquest could bring.
The second reason is political culture and this argument has existed since Polybius. Rome's government and military were very different from Hellenistic governments and militaries. Their government was recognized as an extension of the public body, the Senate AND People of Rome. Therefore it did not need legitimacy from the people because it was elected every year. The military was levied from the population who viewed themselves as actors and beneficiaries of the government instead of mercenary troops. When a Roman army was defeated it did not affect the legitimacy of the Roman government because the Roman government received legitimacy from public participation, not the actions it did. When Hellenistic armies were defeated it proved the illegitimacy of the king because the king needed to prove legitimacy through HIS actions not the actions of the public. Therefore the stakes in these wars were much higher for the Hellenistic monarchs than they were for Rome and that allowed Roman armies to flourish. They could handle multiple (but not infinite) defeats without losing support but if they were able to defeat the Hellenistic armies even once it could be a major blow.
There is of course a strategic and tactical aspect to Rome's success as well, but I think the political and diplomatic aspect is even more important.
Arguably the most important factor which allowed Rome to generally expand in any direction against any foe (especially after its might was unchallenged in the western Mediterranean after the Second Punic War) was its Italian allies.
Some time after Rome defeated its Latin neighbours at the Battle of Lake Regilius in 496BC, it established a network of alliances with other cities in the Italian peninsula, whereby the allies would provide troops under Rome's command whenever the Republic would go to war, receiving in exchange an equal share of any booty, on top of Rome's protection. Livy and Polybius often mention how important Rome's allies were. They provided the manpower which eventually tipped the balance of the Second Punic War (after Carthage was no longer able to support its mercenary armies).
Another important reason why Rome was able to outmatch the Greek world in warfare was its military tactics. Once again, the Second Punic War is important in this regard, since the experience of fighting Carthaginian phalanx units proved useful in Rome's expansion in the east. Scipio Africanus redesigned the Roman legion to be smaller, and more flexible, to outmanoeuvre the extremely rigid phalanx formation. This is clearly seen in the conquests of Titus Quinctius Flamininus against Macedon. The Battle of Cynoscephalae shows how the legion was able to quickly manoeuvre to outflank the phalanx contingents under Philip V of Macedon, which were unable to reposition themselves in time.
If I recall correctly, David Gwynn answers your question in his book, The Roman Republic: A Very Short Introduction. Along with Michael Crawford's The Roman Republic, they form the perfect introduction to the history of the Roman Republic, highly recommend them!