tl;dr - The simple answer is the Athenians dedicated themselves to perfecting naval tactics and technology as opposed to most other city-states, thanks in part to the massive tributes paid to Athens via the Delian League/Athenian Empire which paid for the rowers and naval upkeep.

The long answer really would require me writing a book, but I'll do my best.

When the Athenians discovered a new vein of silver in their silver mines, it was Themistocles who convinced the democracy to use that silver to build such a large navy of trireme ships, claiming Athens could use it against their rivals in Aegina (an island city-state near Athens with a large navy). Triremes were, up to then, the most advanced warship in the Mediterranean. Most Greek navies at the time consisted of slower, less advanced ships. They also tended to be much smaller than what Athens was building up in part because the Greeks tended to view land battles via hoplites and the phalanx as the more honorable form of combat.

Fast-forward a few years later, and Athens now has a large, impressive, advanced navy. They use it against the invading Persians and, with other allied Greek ships, defeated the Persians at the Battle of Salamis. How large was the Athenian navy compared to the other city-states? As one example, roughly half the allied Greek navy at Salamis were Athenian ships.

While today we know that once the Persians were defeated and left Greece they never came back, the Greeks were collectively worried that the Persians might return, most especially the Ionian Greek cities on the coast of Asia Minor (what is today Turkey.) They first asked Sparta to ally with them; however, Sparta wasn't interested in continuing the war. Athens was eager to both continue the war and lead it, and so a new alliance was set up between Athens, the Ionian Greek cities, and the many island cities in the Aegean Sea (roughly two hundred members at any time). Today we call that alliance The Delian League, as it met on the island of Delos. The Athenians led, and the other members could pay for the expenses of the League either with ships of their own navies, with men (rowers) or with money. The vast majority of members chose to contribute to the League with money, which ended up enriching the Athenians and making it easier to maintain a large fleet and pay the rowers who were Athenian. This meant that, as time went on, the navy of the Delian League (which, through the actions of Athens' allies, really meant the navy of Athens) was much more practiced and powerful than any other navy either in Greece or the Mediterranean. As Thucydides puts it:

"Of all the causes of defection [allies attempting to defect from the Delian League], that connected with arrears of tribute and vessels, and with failure of service, was the chief; for the Athenians were very severe and exacting, and made themselves offensive by applying the screw of necessity to men who were not used to and in fact not disposed for any continuous labor. In some other respects the Athenians were not the old popular rulers they had been at first; and if they had more than their fair share of service, it was correspondingly easy for them to reduce any that tried to leave the confederacy. The Athenians also arranged for the other members of the league to pay its share of the expense in money instead of in ships and men, and for this the subject city-states had themselves to blame, their wish to get out of giving service making most leave their homes. Thus, while Athens was increasing her navy with the funds they contributed, a revolt always found itself without enough resources or experienced leaders for war."

Navies are costly; you have to pay for building the ships and maintaining them. You have to pay to train the rowers and pay them a salary; the Hollywood image of slaves chained to rows of oars wasn't true in Greece. Since the Delian League didn't bother to add to Athens' navy with their own ships or send their own citizens to row but just sent money to Athens, Athens used the money to pay their own citizens to row and paid for their own ships. Eventually, the Delian League ended up being more like an Athenian Empire as Athens began demanding higher amounts of tribute and the other members not having the means to stop them. With vast amounts of money coming in, the Athenians could offer high wages to rowers, higher than any other city-state thus ensuring the Athenian navy had the best of the best, so to speak. Furthermore, the Athenians could afford to not only build far more ships than anyone else, but could entice the best shipwrights, the best ropemakers, etc to come to Athens because they had the money.

No other city-state could match the Athenians' resources to make, maintain, and pay for, such a vast navy. The Athenians would also constantly train, coming up with better tactics and perfecting them. They had to because their entire way of life was dependent on collecting the tribute from the Delian League. If another city-state, whether they were in the Delian League or not, were to attempt to become as powerful as Athens on the water, they could potentially interrupt the flow of money to Athens. So the Athenians simply had to be the best on the seas, and could brook no rival, else they would collapse.

Now keep that system going for sixty years. This is why at the start of the Peloponnesian War, and throughout most of it, the Athenians were far and away the masters of the sea. It took the resources of the Persian Empire bankrolling the Spartans to create a navy that could defeat Athens, and even then the Athenians still won victories for years until they were finally defeated.

There are a couple of interrelated reasons. The first is that Athens had long been powerful at sea. The city had many natural advantages on this point, being a wealthy trading hub with territory bordering on the sea on 3 sides, and with 3 good natural harbours right near its urban centre. Many textbooks will tell you that the Athenians first became a contender at sea in 483 BC, when Themistokles persuaded the Assembly to use the silver from a new vein at the mines of Laurion to build a fleet of 200 triremes. However, Hans van Wees has recently argued (in Ships and Silver, Taxes and Tribute (2013)) that Athens already had a fleet of at least 48 ships in the Archaic period (probably the smaller pentekonters), and that this had been expanded to a fleet of probably about 100 triremes by Themistokles' time. What Themistokles did, then, was double the fleet, rather than create it from scratch. He did not revolutionize the character of Athenian power, but merely led the Athenians further down a path they had chosen long ago.

With 200 triremes, the Athenians provided the lion's share of the Greek fleet that fought the Persians at Artemision and Salamis in 480 BC. After the Persian invading force had been defeated, the Greek alliance founded the Delian League with the intention of avenging the losses and sacrilege inflicted by the Persians. The operations of the League were mostly naval, assaulting Persian strongholds in the Aegean and Asia Minor; as the supplier of the largest naval force, the Athenians soon emerged as its leader. Within a few years, its policies as leader of the League took a darker turn. Athens began to regard the League as its personal network of unequal allies, who were compelled by force to remain part of it, and made to follow wherever Athens led.

One of the ways Athens made this shift palatable to the members of the League was by offering them a way out of its constant campaigning: instead of supplying ships and troops, members could opt to pay tribute, which would allow Athens to fund additional naval forces of its own. The result was that Athens constantly practiced its naval expertise and constantly expanded its navy, while the other members of the League increasingly gave up their fleets and lost their expertise, relying on Athens to protect them if the need arose. Any members who rebelled were forced back into the fold and generally made to surrender any ships they had left. By the time of the Peloponnesian War, only Lesbos and Chios still supplied ships of their own. All other subjects opted to pay. Athens now boasted a fleet of over 300 triremes; its sailors had decades of naval campaigning under their belt.

Another major reason is the fact that the most powerful city-state in Greece - Sparta - consciously decided not to try to dominate the sea. This was a recent policy. In the final half of the 6th century BC, we see Sparta operating all over the Aegean and beyond, projecting its power and making a name for itself as a protector of Greek communities against tyrants. To name an example, Herodotos (3.54-55) describes their siege of Samos in 525 BC, which would have required a substantial fleet and expeditionary force. However, in the 5th century BC, Sparta turned inward. They willingly surrendered the leadership of the Delian League to Athens just one year after the defeat of Xerxes' army; after that, they are not seen operating anywhere outside the Greek mainland until the Peloponnesian War forced them to.

There are various possible reasons for this shift in Spartan policy. Later Greek authors much admired them for their decision not to build a strong navy, because this had a noted effect on the political situation at Athens; as the city relied on its navy for defence, the poor who manned the triremes demanded an increasing influence on government policy. Since the Spartan state was built for stability, the Spartans may have been afraid of what would happen if they chose to go down the same route as Athens. Another reason could simply be that Sparta became increasingly conservative as its famous laws took shape, and increasingly wary of overseas adventures that might expose Spartans to alternative ways of life. The Spartan regent Pausanias was only abroad for one year, leading the Delian League against Persian holdings in the Hellespont, before he was accused of taking on Persian airs and sympathising with the enemy. Finally, there are signs that the Spartans were getting increasingly nervous about a helot revolt at home, which indeed broke out after a devastating earthquake in 465 BC. They apparently decided they needed their men to stay around.

Athenian naval power, then, was born out of the combination of aggressive Athenian policy to establish naval supremacy and Spartan reluctance to challenge them over it. Their eventual technical and tactical superiority came from the prolonged naval experience which their very rise to power had denied the other Greeks.

This is not to say, though, that no other Greeks could have given Athens a run for its money. The Spartans had few ships, but they could rely on their subject ally Corinth to provide triremes and naval expertise; Corinth was busily building its own network of dependencies in Western and Northern Greece through much of the 5th century BC. They were able to launch more than 90 triremes of their own. The inevitable clash between Corinth and Athens happened first when Athens destroyed its old naval rival Aigina in 458 BC (recognising the threat from this island right outside the port of Athens), and then when the strong naval power of Kerkyra (Corcyra) went to war with Corinth and made an alliance with Athens. The Athenians were delighted to see Kerkyra's 110 triremes join their side, knowing that a combination of Corinth and Kerkyra could have made for a serious rival at sea (Thuc. 1.36.3). Finally, there was Syracuse - a wealthy trading port like Athens, able to launch a large fleet of triremes. It was against Syracuse that the Athenian fleet first suffered serious losses. Despite significant reinforcements, the Sicilian Expedition ended in total disaster for Athens, with nearly half the Athenian fleet sunk or captured by the Syracusans. At that point even Sparta decided it was worth trying to compete with Athens on the sea.

Edit: beaten by ten minutes. Well, I hope there's still something of value in this duplicate :)