We don't always know exactly where a piece of Athenian legislation came from, but in the case of ostracism, the Athenaion Politeia makes it pretty clear. This text (the so-called "Constitution of the Athenians") is attributed to Aristotle, but is actually the work of one of his students, who compiled the political history of various Greek city-states for him to use as sources for his works on the ideal political system. The essay on Athens is the only one that survives. It was not part of any manuscript tradition, but was discovered on scrap papyrus in a rubbish dump in Oxyrhynchus in 1879. The text is datable to the late 330s BC.

In Ath.Pol. 22.1, ostracism is grouped with the reforms of Kleisthenes, which means it was introduced in 508/7 BC:

These reforms made the constitution much more democratic than that of Solon; for it had come about that the tyranny had obliterated the laws of Solon by disuse, and Kleisthenes, aiming at the multitude, had instituted other new ones, including the enactment of the law about ostracism.

The main point made here with regard to ostracism is that it was intended as a measure to soothe or favour "the multitude" (ho plethos), that is, the common people. Kleisthenes is generally regarded as the founder of Athenian democracy, but this may be giving him too much credit; the degree to which fifth-century Athens was democratic is debatable. Rather, his measures were intended to break up the old power bases of factions within the elite, and to reorganise Athenian political institutions along less partisan lines. All this was intended to prevent Athens falling back into tyranny. In this context, ostracism was the ultimate weapon of the people against would-be tyrants: whoever seemed to be striving for one-man rule could be exiled at the discretion of the Assembly. The law prevented the bloodshed that usually came with a grab for supreme power or its prevention.

In principle, ostracism was extremely democratic, in the sense that it legally placed the Assembly above any single citizen. Whatever power any single member of the elite was able to accumulate, the people would always be able to gather in the Assembly and decide to exile him. However, in practice, it seems the Athenians were hesitant to put the new law to use. According to Ath.Pol. 22.3, it took 20 years before the first official ostracism:

In the archonship of Phainippos, two years after the victory [at Marathon], the people being now in high courage, they put in force for the first time the law about ostracism.

Presumably, the people's disinclination to ostracise members of the elite was due to fear of repercussions. Early Classical Athenians had grown up in a world where elites wielded a tremendous private capacity for violence; just two generations earlier, the tyrant Peisistratos had gathered enough of a personal retinue through money and connections to defeat the army of Athens and submit the state to his will. When the Peisistratids were finally driven out, Isagoras nearly managed to seize the tyranny for himself with Spartan aid. The wealth and influence of the leisure class was staggering, and it was difficult to imagine a world in which these ultra-competitive "people-eaters" would simply bow to the will of the Assembly. It took a while for Athens to grow confident that its new constitution was strong, and that its militia could take all comers.

It made sense to start with people who were unlikely to be able to count on broad support. At first, the Assembly went after targets that were widely regarded as suspicious - mostly members of the elite network of the Peisistratids. Once they were assured that the system worked, they became bolder:

For three years they went on ostracizing the friends of the tyrants, at whom the legislation had been aimed, but afterwards in the fourth year it was also used to remove any other person who seemed to be too great; the first person unconnected with the tyranny to be ostracized was Xanthippos son of Ariphron.

After this initial rash of ostracisms, though, uses of the law quickly became rare. Because ostracism leaves very recognisable traces in the archaeological record (in the form of pot shards with names on them), it is relatively easy to be sure that we aren't missing anyone due to any silence of our sources. The number of pots with the name Megakles suggest that he may have been ostracised twice, but other than that, we have a pretty good idea of all the people who were ever ostracised. There are a few stray examples from the 470s to the 440s (including famously Themistokles, Kimon, and Thoukydides son of Melesias (not the famous Thucydides)), but at the end of that period, Perikles was firmly in charge of Athenian affairs. There was probably little reason to fear the rise of another tyrant. From 440 onward here is no example of ostracism at all until the isolated and strange case of Hyperbolos in 417/6.

This case is described in some detail in Plutarch, Life of Nikias 11. Essentially, two opposing factions in the city proposed a competitive ostracism vote: either Nikias or Alkibiades was to be exiled. Since neither was keen to take the verdict lying down, the two candidates for exile worked together to raise a third: Hyperbolos, a demagogue who is despised in all our sources. Hyperbolos managed to deflect enough votes from Nikias and Alkibiades to "win" the ostracism competition. The other two got to stay, with disastrous results for Athens.

According to Plutarch, the clear manipulation of this case of ostracism discredited the entire law in the eyes of the Athenians. Furthermore, their initial use of this law against great commanders like Aristeides, Themistokles and Kimon made them feel like it was almost an honour to ostracise lesser men, making them reluctant to use the law when it was needed. Whether this really caused the abandonment of ostracism is up for debate. It is also possible to argue that the democracy really only came into its own after the death of Perikles, and that the lack of ostracisms is a sign that its curtailing and redirection of the competitive energy of the elite was successful. In Athens, even the richest simply weren't aiming for tyranny anymore. However this may be, Hyperbolos is the last known case of ostracism. Throughout the remaining century of Athenian democracy, 416-322 BC, it was never used.

Further, did any other polities adopt its use from the Athenians?

To my knowledge, ostracism was an exclusively Athenian institution.