I have always heard the founders didn't want the capital to be apart of a state but never explained why. What benefits did they feel this gave?
At the time of the creation of Washington D.C., the government did not control the military, and the states provided militia to keep the peace. An 18th century National Guard in a way.
In Pennsylvania, in 1783, the Congress of the Confederation(the federal governing body of the period), was surrounded by 400 soldiers of the Continental Army demanding pay that was owed to them. It was in effect, a union strike, except that the strikers were soldiers and armed. When Congress asked the Pennsylvania Council(state government), to mobilize the militia to protect them, the Council refused.
This gave Congress a problem: they had been unable to rely on the state to protect them, and could not trust that the states would provide protection in future. With this realized, they planned a federal district, independent from the States and their governance, where Congress would have complete legal control and charge of their security.