The straightforward explanation for NATOs involvement is that NATO was written into the Dayton peace treaty and mandated by the U.N.to lead the implementation force (ifor) into Bosnia. But I'm guessing you mean more why NATO, what could they do that the U.N. couldn't.

The answer is a lot. Unprofor, the U.N. mission prior to ifor was seen as a major failure, with too few troops, poor political leadership and unclear rules of engagement. Massacres such as Srebrenica happened with U.N. troops nearby, because they lacked the manpower, weapons and high level support they would have needed to intervene. One way to look at international orgs is that they are tools. Countries use them to do things they could not accomplish alone. NATO, with integrated command and control, a set chain of command and a relatively fast (for international orgs) political decision making process, is much more responsive and flexible in a military operation than the U.N.

NATO was also, quite frankly, searching for a purpose at that point in history. And the violence from the breakup of Yugoslavia worried western European nations, who had already been surprised by the fall of communism and breakup of the ussr. NATO as an alliance was ready to prove they had a positive post cold war role to play, especially in Europe.