In order to understand how red dot sights work, you must first understand why traditional rifle scopes work the way they do.
In normal rifle scopes, there are several lenses that produce the final image that you see when looking through the scope. These lenses bend the image into a cone of light. The cone of light that produces the image you see is concentrated on a single spot behind the scope.
Because of these lenses, you must have your eye at the correct position to get a proper view through the scope. This correct position is referred to as eye relief. For example, when a rifle scope's specifications say eye relief is four inches, that means that your eye must be exactly four inches away from the lens closest to your eye for optimal viewing. Correct eye relief will allow you to see a perfect sight picture without clipping (black crescents of light or a black ring around the sight picture).
In addition to eye relief, there is also a term known as exit pupil. Exit pupil is the size of the cone of light coming from the scope at the correct eye relief. If you have correct eye relief but your eye falls outside this area of exit pupil, you may still see clipping. It's slightly more complicated than that, but it all boils down to this: Your eye has to be at the correct position to ensure you have a perfect sight picture.
And that's where red dots are different.
Red dot sights project an illuminated reticle on to a reflective element on a single lens. Once the red dot sight is attached to a firearm and sighted in, that dot will always point to where your bullet is going to go.
Red dot sights do not typically utilize more than one lens element, and do not magnify the sight picture. Because of this, there is no eye relief or exit pupil. No matter where your eye is in relation to the sight, your bullet will go wherever that reticle is pointing.
Red dot sights are commonly used in close quarters because they allow the shooter to quickly and accurately sight a target. In military and law enforcement situations, this quick and accurate targeting can mean the difference between life and death. For hunters, it means the ability to get the jump on game at close range.