The term "anarcho-capitalism" has, we might say, rather an arresting quality. But while the term itself may jolt the newcomer, the ideas it embodies are compelling and attractive, and represent the culmination of a long development of thought.
If I had to boil it down to a handful of insights, they would be these: (1) each human being, to use John Locke's formulation, "has a property in his own person"; (2) there ought to be a single moral code binding all people, whether they are employed by the State or not; and (3) society can run itself without central direction.
From the original property one enjoys in his own person we can derive individual rights, including property rights. When taken to its proper Rothbardian conclusion, this insight actually invalidates the State, since the State functions and survives on the basis of systematic violation of individual rights. Were it not to do so, it would cease to be the State.
In violating individual rights, the State tries to claim exemption from the moral laws we take for granted in all other areas of life. What would be called theft if carried out by a private individual is taxation for the State. What would be called kidnapping is the military draft for the State. What would be called mass murder for anyone else is war for the State. In each case, the State gets away with moral enormities because the public has been conditioned to believe that the State is a law unto itself, and can't be held to the same moral standards we apply to ourselves.