How could the small Greek city-states defeat the huge Persian Empire?

They banded together as a cohesive force, producing a combined navy which outmanoeuvred and defeated the Persian navy; and then without the threat of naval attacks on their cities, they were able so send out their armies to combine and defeat the Persian army. The Persian army was also weakened as, after losing the ability of their navy to protect their resupply fleet coming from Asia, they then had to send half their army home as they could not feed it, and this evened up the size of the opposing armies. And the Greek army had armoured warriors which outclassed the unarmoured soldiers of the Persian army.

It all hung on the elimination of the Persian fleet at Salamis.