Final answer:
According to Thucydides, Athens became the "hegemon" of the Delian League due to the fact that Pausanias, the Spartan King (or, at least, regent), was recalled by the Spartans to face charges of treason. The Ionian Greeks suggested that the Athenians should take the leadership, and Athens accepted. Sparta then relinquished the leadership due to its domestic problems and goodwill towards Athens at the time.
Step-by-step explanation:
According to Thucydides, Athens became the "hegemon" of the Delian League due to the fact that Pausanias, the Spartan King (or, at least, regent) who had been initially placed in command of the allied forces, was recalled by the Spartans to face charges of treason. The Ionian Greeks made overtures to the Athenians suggesting that they should take the leadership due to their common descent, high prestige after the Persian Wars, and large and experienced navy. Athens accepted and, as such, when the Spartans sent out another commanding officer, Dorcis, he was rejected by the Ionians. Sparta, due to its domestic problems and general feeling of goodwill towards Athens at the time, was content to relinquish the leadership.