The simplest explanation I have ever seen is by a Gem Singer, a member over on the LF pages.
http://www.largeformatphotography.info/forum/showpost.php?p=531354&postcount=2
After all the charts and graphs and chest pounding, this simple explanation really brings the fundamentals into reach of the average photographer who wants to take better pictures. Once you have this concept under your belt, you can get into the whole libraries of books written on the subject.
I've quoted his whole post below...
Aim the one degree spot at the darkest area in the scene where you still want to see some detail. Close down two stops.
You have just placed the shadow area in Zone III.
That's usually the proper exposure for the scene.
Now, aim the spot at the brightest area in the scene.
If it's a five stop range between the darkest and brightest reading, use normal development.
Less than five, increase development.
More than five, decrease development.
No need to take a whole series of meter readings and average them. That's defeating the purpose.
Just make certain that you have given enough exposure to get some detail in the shadows. Then, develop for the highlights.
That's the Zone System in a nut shell.
Cheers,