Well, the most basic answer is that Artemisia was the Tyrant-regent of Halicarnassus, and she sided with Xerxes because she was a Persian subject and his subordinate. She gets a lot of attention because she was semi-unique as a prominent female naval commander (though interestingly enough, another Artemisia from the family of the Tyrant of Halicarnassus was also a naval commander 130 years later). She's also notable because she received a lot of praise and good press from Herodotus in his Histories. This is partly due to the fact that they were from the same city and possibly distant relative, and partly because she was a Greek in close proximity to the Persian king and thus an excellent story-telling device for Herodotus' style. Herodotus, like most ancient historians, added dialogue, speeches, and actions he couldn't possibly have known about to move the narrative, make a moral message, or make the Greeks more important to Persian history, but he very rarely seems to have just made up actual people.Despite being notable and relatively well known even today, Artemisia's mere presence on the Persian side is not at all remarkable.
Halicarnassus had hegemony over the nearby islands of Kos, Nisyros and Kalymnos, and was the pre-eminent city in the region of Caria, roughly the southwest corner of modern Turkey. Caria was home to several Greek cities that had been founded sometime between 1200-800 BCE, which all spoke the Doric dialect of Greek. Following the coastline north, toward the Hellespont, there were many more Greek cities founded in the same time frame, which spoke the Ionian and Aeolic dialects of Greek. These were cities like Miletus, Ephesus, and Smyrna. Around 540 BCE, the Persians, under Cyrus the Great, conquered these cities and brought them into the Persian Empire. Many of them tried to secede from the Empire during the Ionian Revolt of the 490s, but the Revolt failed and Persian rule was re-established.
The Persians ruled the city through a Greek system called "tyrannies." That word didn't have the negative context that it has in modern English for the Greeks. A tyrant (Greek: Tyrannos) was just a Greek autocratic ruler who didn't have any traditional legal rights to their power. In the 8th-6th centuries BCE, these became very common in Greek city-states, and the Persians propped up tyrants of their choosing in their Greek cities. The tyrants reported to the governor (satrap) of the province of Lydia, who in turn reported to the king.
Artemisia's father Lygdamis was the Tyrant of Halicarnassus. When Lygdamis died, his son Pisindelis was still a minor, so Artemisia ruled in the meanwhile. As subjects of the Persian Empire, Halicarnassus and like all Persian subject cities, furnished troops and ships when Xerxes called on them to wage a war against the European Greek cities on the mainland. Evidently, Artemisia was fiercely independent Greek woman and intelligent strategist because she led her fleet herself rather than sending one of her subordinates.
While some other Greeks switched sides and betrayed the Persians, Artemisia had no desire to do this. In general, the Tyrants of the Persian-Greek cities were not very popular and relied on their relationship to the Persians to maintain their positions of power. For Artemisia, this was doubly true because she derived all of her power from being the sister and daughter of official Tyrants and could not truly seize power in her own right.
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