The reason is that the flux from any particular star goes as 1 / distance^2, but the number of stars observed in a patch of sky increases like the distance^2. The two cancel each other out.

An easier way of thinking of it is to think about looking along a single direction. No matter which direction you look, this ray will always be pointed towards the surface of a star.

(Incidentally, when thinking about these sorts of problems, it helps to distinguish flux from specific intensity: http://en.wikipedia.org/wiki/Specific_radiative_intensity)