Why isn't the cosmic microwave background a solid 3D volume? This is the cosmic microwave background radiation:

It's a Mollweide Projection, which maps the surface of a sphere. Like, this is a map of Earth (also Mollweide Projection):

And it only shows the outer surface of Earth. You can't see the inside.
What I don't get is, how can the cosmic microwave background radiation be mapped as a flat sphere? The CMB is everywhere in the universe, so why isn't it a solid 3D volume? How come it doesn't look like this:

Why is the CMB a sphere as opposed to a volume?
 A: It is a solid volume, but we can't see most of it.
The plasma that emitted the CMBR is/was located on a spacelike 3D surface. Everything that we can see lies on the boundary of our past light cone, which is a 3D null surface. The only part of the CMBR that we can see is the intersection of those two surfaces, which is a 2-sphere.
(Actually, the CMBR surface has a nonzero thickness because the plasma has an optical depth, and the light cone has a nonzero thickness because the universe isn't a perfect vacuum, so the intersection is a "thick sphere". I don't know the thickness, but this paper, if I'm interpreting it correctly, estimates it at ~19 Mpc in present-day comoving distance, or ~0.1% of the 14 Gpc radius of the sphere. This is comparable to the angular resolution of Planck, so Planck's CMBR map may be the best we'll get for the next ~19 Mpc/c.)
A civilization at a different location in spacetime will see a different spherical slice of the CMBR. If they're close enough to us, the slices will have a circle in common. This is the intuition behind Cornish et al's search for correlated circles.
A: This is because the CMB is coming at us, in all directions, from behind the most distant galaxies in the visible universe. In essence, when the source is THAT far away and carries with it no explicit information about exactly how far away it is, we can simply map it as an intensity map on the inside of a sphere of radius ~(extremely large) and leave it at that. 
