Why was the Delian League able to drive the Persians out of Anatolia?

The Delian League was defensive, and protected the Greek city-states, not to take over Anatolia, which remained under Persian control except for the Greek cities. The 180 cities of the Delian League paid for the navies of Athens, Samos, ChiOS and Lesbos which were able to keep the Persian navy away and provide ground support for threatened cities. After 30 years the Persians gave up trying to regain control of the cities and left them to their own devices, agreeing in the Peace of Callias not to go beyond specific lines by sea, and accept the independence of the Greek city-states in Asia Minor.

Athens the Persian threat subsided, Athens converted the Delian League into an empire of its own and used its war contributions for its own benefit, and used the League resources to try to dominate Greece, leading to the Peloponnesian War against the Peloponnesian League led by Sparta. After losing this war by 404 BCE, Athens was stripped of its empire, and became a second rate power. A short time later Persia took the opportunity to reabsorb these cities in Asia Minor and impose the 'King's Peace' on the still-warring Greek city-states in mainland Greece, whose fighting was spilling over into the Persian Empire.