Plenty of other city states had democracies. They arose both independently and because of Athenian intervention. Corcyra is a great example of this. Thucydides writes about the conflict between the aristocrats and the democrats on that island; it is his standard for "stasis" - that is civil war (book 3 I think). Another state that dabbled in democracy was Syracuse (between dictatorships) and Athens generally supported democracies among its empire (or allies, depending how generous you want to be). Just looking at my index in Thucydides I see references to democracy in Argos, Boeotia, Corcyra, Samos, Thasos, and Thespiae. This doesn't even include cities in Italy or Asia Minor.
I'm sorry to say it but I think the premise of your question is flawed.
While it's true that Athens was by no means the only classical Greek city state with a democracy, there were many that remained an oligarchy (e.g. Sparta, Corinth), so I think this question can still be answered.
Originally all Archaic Greek city states were oligarchies or monarchies, simply because that was tradition since the Mycenean period and Bronze age. In 594, the Athenians were experiencing massive civil unrest because of poor economic situation, clan rivalries, and a practice called debt-slavery, where the poor became owned by the rich if they couldn't pay their debts [1]. This situation caused the Eupatridae (the aristocracy) to elect Solon as the archon (head of state). Solon saw the need to reconcile the rich and poor, so abolished debt slavery, opened the assembly to all Athenians, including the poor, and generally set the stage for true democracy (which arguably didn't happen until Cleisthenes' reforms, 86 years later)
Generally, other Greek city states become democratic because of the influence of Athens, either directly, where Athens forcibly instated democracy (it was kind of like the America of Ancient Greece), or indirectly. Athens could do this because it was the leader of the Hellenic League (later called the Delian league), a Greek alliance against Persia. Once the war had finished, Corinth and Sparta left the league, so were basically immune to the influence of the Athenian empire.
As for its success, that's a very subjective question. Athens had immense power and wealth during its golden age, and is remembered for its art, philosophy, and of course government. However Athens was ultimately defeated by Sparta in the Peloponesian Wars, so it could be argued that oligarchy was the more successful form of government.