How can it be that we can still see the CMBR?

[MigL]

The best way to think of the CMB is not as linearly emitted radiation.

!3 billion years ago the universe was at a temperature where electrons couldn't stick to atoms. It was essentially a plasma much like our sun.
At this characteristic temperature, a black body radiates ( EVERYWHERE ), centered in the visible light range of the EM spectrum.

The universe has since expanded more than three orders of magnitude; and as the natural gas laws predict, it has cooled proportionately to 2.7 degrees. At this characteristic temperature, a black body also radiates ( EVERYWHERE ), but at much reduced frequency/increased wavelength. The peak of the radiation curve is now centered in the microwave region.

IE, the CMB radiation wasn't emitted 13 billion years ago.
It is being emitted NOW, by a black body at 2.7 degrees.
That is why we see it everywhere.