Question

We often say that the cosmic microwave background radiation (CMB, or CBR) is a direct viewing of the Big Bang. But actually, the temperature of this black-body radiation (3 degrees K) is insanely low for the Big Bang. Why is it so low, when the Universe started out so hot? And why do we say now that the CMB represents light from about 400,000 years after the Big Bang, rather than the Big Bang itself?

Answer 1

Up until, oh, about 400,000 years after the Big Bang, the universe was opaque or foggy. So the photons bounced around a lot. It was only after the universe decoupled that the light was able to go anywhere significant, and by this time, the universe was getting a lot bigger.

There's a similar but less severe thing going on in our own Sun. If some hydrogen fuses and emits light, it takes quite some time for the light to get out of the Sun. How much? I've heard a million years, but I don't know if that's right. In any event, it's much longer than it takes the light to get to us once it's out.

Now, as to the speed of the expansion of the universe, if you put the age of the universe together with the measured size, you get a size in light years much bigger than the age in years. That's puzzling. However, it's a rather naïve calculation, making Galilean assumptions. Fortunately, we have General Relativity. Unfortunately, it has an awful lot of little numbers in it.
Sometimes people say that space-time was expanding faster than light, but that's also naïve and somewhere between highly misleading and flat-out wrong. General Relativity is purely local, and taking that plus the idea that it works everywhere (and everywhen), you get the right answer. Things that happen at a distance, however, seem weird and counterintuitive.