Q:

Why did the USA support Kosovo in the war against Serbia?

Wikipedia has quite an extensive article about the war in Kosovo, but it doesn't do a good job at explaining the rationale of the US/NATO joining the war. What was the official reason for escalating the conflict to an international war and what did the US gain from intervening?

A:

I do not know if later studies have shown otherwise, but at the time the reason given was that the Kosovo conflict was a continuation of the wars that happened after the breakup of Yugoslavia.
The Bosnian War had raged from 1992 to 1995, and had stopped only after the NATO led an air campaign against the Serbian backed faction, and that only after a more moderate international intervention by the United Nation peacekeeping forces failed to stop the fighting and the massacres.
With that memory still fresh, it is not surprising that when news appeared that there was again fighting between separatists and Serbian military forces, and civilian massacres the same mechanism was put in motion1
As for why did the NATO have to take part into it, this is more opinionated but I think I am not very mistaken if I claim that:

After the failure of other measures to stop the war and atrocities in Bosnia, there was public pressure (specially in Western Europe countries) to avoid a repeat of it in Kosovo.
It was a time shortly after the fall of the SU that lead to the policies of "New World Order"2 which saw increased USA interventionism ("the World police").

Probably, the fact that Serbia was politically supported by Russia and that Russia was in its lowest hours and mired with economical and internal problems could have been a factor, too.

1 Although politically the situation was not exactly the same (Kosovo was a province of Serbia, while Bosnia and Croatia had been separated entities), and initially it was intended that Kosovo would remain part of Serbia and that the objective of the war was only to force to stop the fighting.
2 And that famous "The End of History".

