In about 483 B.C. the Athenians found new reserves of silver at the mines of Laurion. The Athenian general Themistocles persuded the Athenian assembly to allocate the expected income from the new mines to expanding the Athenian fleet. The fleet was expanded to 200 ships which served as the base for their Naval hegemony in the years to come. Their navy served a vital role during the second Greco-Persian War at both the Artemisium Straits and at Salamis. Following the Greek victory over the Persians, Athens convinced other Greek city-states to join the Delian League as an Alliance to guard against another Persian attack. Members of this league would offer either ships or money to Athens who oversaw the defense of the Aegean. The money that most of the city-states offered (the vast majority of the ships in the league were Athenian) was sent to a treasury at Delos which is where the name of the league comes from. The Athenians used this money to further build up their navy and to attack and take over Persian territory along the eastern Aegean and the coast of Asia Minor. This navy allowed them to secure trade routes from Egypt all the way to the Black Sea as well which enhanced their economic strength beyond the extent that of the tribute went.

The other answers here are useful but ultimately don't go back far enough. Laurion and the Delian League were both important parts of Athenian power, but it had been a major power and uniquely large city well before that. The reason probably comes down to political institutions--very early on Athens developed a very strong sense of local identity that encompassed the whole of Attica. This meant it had a much larger and more unified citizenry than other poleis. As importantly, it was well situated commercially and became an important port early on (and indeed, Athens is still the busiest port in the Mediterranean).