Why was the battle of salamis so important?

After failing to defeat the Persian fleet at Artemesion opposite Thermopylai, the Greek fleet moved south and regrouped at Salamis near Athens for another try. They had to defeat the Persian fleet as it threatened the southern Greek cities, who kept their armies at home to protect them.

When the Persian fleet was defeated at Salamis, the Persians faced a winter in a poor country, and as they could no longer protect at sea their supply fleets coming from Asia Minor, they had to send half their army home as they couldn't feed them.

The following spring, with the maritime threat gone, the southern Greek cities sent out their armies to join up and defeat the reduced Persian army at Plataia. At the same time, the Greek fleet finished of the remainder of the Persian fleet defeated at Salamis where it was holed up at Mykale.

Defeat of the Persian fleet was the key to the whole Greek strategy, and success at Salamis implemented it.