How did the Greeks defeat the Persian army?

They used an effective military formation known as a phalanx.

They surprised the Persians by crossing the Aegean Sea to attack.

They had many more soldiers than the Persians did.

They captured the Persian emperor in a raid on his headquarters.

The Persians had a long supply line that was vulnerable to attacks. Having defeated the Persian fleet at the Battle of Salamis the Greeks gained the mastery of the seas, and this allowed them to harass and completely cut off the Persian supply lines.

Once their supplies lines were cut, the Persian numerical superiority turned into a liability.