How were the Greek armies able to defeat the much larger Persian forces in the Persian Wars?
In general, it appears that the Greeks were able to defeat the Persians because of their superior battle tactics. Of course, we must remember that the victors write the history books and that most of what we know about the battles of the Persian Wars come from Greek sources. What we know, though, seems to indicate that the Greeks were better led and that the Greek phalanxes were a superior way to organize for battle.
We can see the Greeks' tactical superiority in battles like that at Marathon and at Salamis. In both of these cases, the Greeks disposed their forces better and were also able to lure the Persians into doing what they (the Greeks) wanted. At Marathon, the Persian attack on the weakened Greek center allowed the strengthened Greek flanks to attack and essentially encircle the Persians. At Salamis, the Greeks manipulated the Persians into attacking in narrow waters where the Greeks' heavy ships with hoplites aboard could negate the speed and manuverability advantage of the Persians.
In these ways, what we know of these long-ago battles indicates that the Greeks won through superior tactics.