Reading up on the Achaemenid Empire, i realized just how vast it was - according to Wikipedia it is believed to have encompassed up to ~44% of the world population at the time too. In addition to that it seems to have had a highly organized central power (roads, postal system etc).
How did the greek city states possibly manage to resist this empire? Were the Persians fighting other wars parallel to the Greco-Persian wars?
Was it resistance of the local population in Asia Minor/Anatolia making it impossible for them to fight effectively?
Were there Internal struggles within the Persian Empire?
Edit: I'm aware of Thermoplyae to a degree, but what happened before and after that? The Persians seem to have struggled with the Greek quite a bit.
Part of it was technology, part of it was discipline and tactics. The Greeks knew their own land better than any Persian and they had bronze weapons. Greek hoplites could cut through Persian infantry like butter because the Persians use wicker shields and wore cloth or leather in terms of armour.
In terms of land usage you only need to consider the battle of Thermopylae where Leonidas held off the bulk of the Persian forces with a significantly smaller force. It wasn't quite the romanticized 300, but it was a much smaller number, likely around 3000 Greeks against 70,000 to 300,000 Persians. Using the land Leonidas and his fellow generals could hold the greater army by funnelling them into strategic positions to reduce the effect of their numbers.
After Thermopylae the Persians controlled most of Greece but the next major Greek victory came at the battle of Salamis where the Greek general Themistocles lured the Persian navy into cramped quarters battling in the narrow straits where, again, the greater numbers didn't give them any advantage.
After the Greeks broke the Persian navy at Salami there were attempts at making peace but ultimately two battles decided the fate of the Persian invasion. The battle of Plataea, where the Greek armies and Persian armies tried to position for their best advantages, resulting in a stalemate, and the battle of Mycale, where the remnants of the Persian fleet were decimated, establishing Greek supremacy in the Aegean.
The battle of Plataea was decided when the Greeks attempted a retreat. The Persian general, thinking the Greeks truly defeated, ordered a charge. The Spartans, Tegeans, and Athenians stopped their retreat to fight instead where they managed to trap the Persian army in its camps, slaughtering them in massive numbers.
It should be noted, however, that the Persians had pushed deep into Greek territory, Athens had been abandoned, and many Greek states did fall. Thrace had come under Persian control for example and Achaea was the only major Greek power that hadn't been at least partially conquered before the tides turned. If it weren't for superior weaponry and tactics the Greeks would not have stood a chance.