How did the Athenians continue to fight after Athens was burned?

They abandoned the city before the arrival of the Persian army, their families seeking shelter in the cities of the Peloponnese. Some males stayed in the city, trying to defend the Acropolis (failed), some located themselves on the island of Psyttalia, but most embarked on the 180 ships of their navy and fought at sea.

Athens was not burned. The Persians looted it and destroyed the statues of their gods in retaliation for the Athenians burning the Persian gods earlier in Sardis. The Persian didn't burn the city as they used it as accommodating their army in 480 and 479 BCE.