The material radiating the CMB is all around us (left). If the universe were not expanding then it would radiate like a 3000K black body, and the radiation field we would be exposed to would be 18 MW/m2 -- the intensity is independent of distance in a static Euclidian space. What renders this particular contribution to the night sky brightness so "dark" is redshift and expansion. At at redshift of 1000 the surface brightness is reduced to 1/(1000)4, which is 18µW/m2 -- or a 3K black body. Cosmic expansion saves us from living inside a 3000K oven.
Figure: Whittle's (upcoming) undergraduate text.