It allows you to shoot without precisely lining anything up. If the dot is on the target you will hit where the dot is, period. Even if the dot isn't centered in the tube/sight window- it's kind of like a laser dot. You shoot with both eyes open, don't have to have your cheek against the stock and don't have to have your head in some spot in relation to the sight or in line with anything.
Go to http://www.aimpoint.com ; most your questions are answered on there. They have a demo and good explanations of how they work.
I have been through the same situation. If you are used to regular scopes and irons it seems a little weird how a good red dot sight works. Hell it seems impossible, but once you play with one you'll find how cool they are and how well they work.
A 4 moa dot is the same size as the width of the front sight post on an AR so you really shouldn't notice any difference in accuracy over irons. The difference is speed- the dot is ungodly fast compared to irons and encourages both eyes open shooting.
Also, you just ignore the sight and act as if it isn't there. Focus on the target and forget the tube or window is even there. Where the dot appears in relation to the tube doesn't matter. The gun hits where the dot is when the trigger is pulled regardless of the dot's position in relationship to the sight.