The counterpart to this question is 'why was Alexander and his army able to conquer the Achaemenid Empire'- Macedon was a much, much smaller kingdom that was a former client state of the Achaemenids whose history generally consisted of fratricidal warfare, fighting petty local kings technically under their authority, and fighting back various non-Greek polities along their borders. But in ten years Alexander and the forces available to him were able to systematically outfight multiple Persian armies and take total command of the Achaemenid system. The reason I bring this up is that these two realities both seem counter-intuitive given how large and prosperous the Achaemenid Empire was. But the fact that both of these two things happened obviously means that something in this picture doesn't fit.

The first thing to reconsider about the situation is talking about the Achaemenids as having an advanced military. In terms of their equipment, at least, all states in and around the Mediterranean in this era would have been peers of one another, there was no great technological disparity between the Achaemenids and the Greek city-states. The Achaemenids had fleets of hundreds of triremes, the default warship of the era, and eventually so did the Greeks. Both used bronze and iron weaponry. The only real difference is that the Achaemenids had far greater experience and mechanisms in place for dealing with the logistics of large armies, which mostly related to its structure and necessity.

It is definitely true that the Achaemenid military was far larger than that of all the Greek city-states combined, for it had access to a far larger population all of whom were united under one administration and king, as opposed to Greek city states and kingdoms who fought one another in ever-shifting alliances. However, this resource was always of limited applicability. There was always a limit as to how much military force the Achaemenids could apply at one time. Whilst the core of the army was made up of professionals, semi-professionals who were also landowners and farmers and smallholders would have been one of the main sources of manpower in a large campaign. In an emergency they might even have to institute a standing levy in given regions. In both cases this represents labour and in a number of cases oversight that is no longer being used for those purposes. In other words, people who aren't harvesting, who aren't in control of arable land, who aren't making things, who aren't doing all the important jobs that these complex societies relied upon. Therefore there was not only a limit as to how many men could feasibly be raised under arms at once, there was also a time limit as to how long those men could reasonably remain under arms. Generally speaking the big royal armies of, say, 200,000 or 250,000 men could only exist for one campaign season, particularly if this was far away from the heart of the Empire. And Greece was nothing if not far away from the heart of the Empire, it's over 2,000 miles between Athens and Babylon. There would have been men under arms as permanent garrisons, frontier troops and similar elsewhere, for sure. But neither could the Achaemenids commit all men under arms, because they had many enormous frontiers that required policing, and they also relied on their garrisons to help ensure law and order was kept.

We also have to take into account that the Greeks allied against the Persians made a number of astute decisions. During the war with Xerxes their combined fleets were well commanded, the Isthmus that led the way so many of the anti-Persian states was fortified leaving the Persians with very few options as how to approach the Peloponnese, the Athenians successfully evacuated Athens leaving their citizenry and armed forces intact, and most important of all they never committed their largest forces to the field until after Xerxes and much of his army had left.

Speaking of forces and size, one other factor to consider is that the Greek forces are often consistently underestimated in size. The main reason for this is that historical accounts of the era tend to exclusively concentrate on the number of hoplites in a given army, because the hoplites in many places were a prestigious social class in their own right and the Archaic-Classical Greek culture tended to focus their prestigious portrayals of warfare on hoplites. But we know that the Greek armed forces of this era did not just consist of hoplites, even if they looked down on those performing other roles. What we are mostly talking about here are light troops and skirmishers; for example, Spartan Helots were expected to serve as light troops on campaign alongside the Spartans. And we must also counterbalance that with consistent over-estimates of Persian numbers in Greek sources- modern estimates of the full Persian army led by Xerxes tend to range from 150,000-250,000 men rather than the million or two million ancient sources give. In addition, the army that Mardonius was left to pacify the new 'province' may actually have been outnumbered by the Greek force assembled at Plataea. This is not total war or full mobilisation, it doesn't really matter if the Persians in theory had 2 million armed men across the Empire if they only had 100,000 in Greece and the Greeks could defeat that specific army.

And, having said all this, the Persians did almost win. The majority of Greek states submitted to or allied with Persia. Between that and the areas taken by force, the Persians occupied around 3/4s of all Greek speaking areas at the height of their control. They had also successfully sacked Athens, even if the Athenian state-in-exile was still totally intact. But Xerxes was always going to have to eventually return home, with the majority of his army, and this could not be delayed in order to see the campaign through. In particular, can you think of the absolute pandamonium that would have struck the Achaemenid Empire if Xerxes had gotten stuck in Greece? Given that the Greeks destroyed much of the remnants of the fleet that survived the battle of Salamis, and could have destroyed the pontoon bridge used to transport Xerxes' army across, that was a very real possibility towards the end of the campaign. The Achaemenids, despite almost winning, did not exist in a world in which they could devote every resource available to finishing the anti-Persian Greeks off, because they were at the extreme edge of the Empire in the first place. It's notable that the Persians were unable to keep any of the territory on the European continent that they conquered under Darius and Xerxes, even the bits that weren't Greek at all. And, as mentioned before, in saving their armed forces from a pitched battle the Greek allies were able to confront the greatly reduced Persian forces with something like equal numbers.

Last but not least, many of those states most opposed to the Persians were some of the most powerful ones in Greece. Coupled with effective decision making, a limit as to how many resources the Achaemenids could commit, how far Greece was from the Achaemenid heartlands, the timetable for the royal army and Xerxes himself, you end up with a picture where an anti-Persian victory was never certain but never quite as unlikely as it might seem.

The one other correction I have to make before finishing is in seeing Greece as a 'country' in this era. It wasn't a unified state of any kind, they regarded one another as fellow Greeks but that didn't stop them warring with one another constantly. You can tell they weren't united by the fact that over half the Greek states in Greece itself sided with the Persians and not against them. The term 'country' makes people think of a unified force, or a modern nation state, when the Greeks as a whole resembled neither.