Why were the Greeks and the Persians fighting in the battle of Thermopylae

Persia invaded peninsular Greece to incorporate it within its empire to stop the incessant wars which spilt over into the empire.

The southern Greek cities resisted, and determined to first destroy the Persian fleet to end the amphibious threat to the Greek cities, and to prevent the Persian army being resupplied by sea.

The pass at Thermopylai was blocked by a small army force from the cities, which the Persians tried to outflank by sea, running into the waiting Greek navy in the nearby strat at Artemesion.

Unfortunately for the Greeks they lost the sea battle, and as the blocking force was no longer required, it was withdrawn. The Spartan and Thespian contingent remained behind to cover its withdrawal, sacrificing themselves to let the others get away.