When CMB radiation was emitted, the universe was only 350,000 years old or so, and so why wasn't it absorbed at that time? The universe is postulated to have been expanding at that point but I think much much less than than the speed of light.
Since the fact that photons were not absorbed by neutral atoms was already addresses, I will address your last sentence, and your comments on the size of the universe and the fact that photons were traveling faster then the speed of the expansion itself.
I understand you are confused because you hear phrases like "the universe was much much smaller at the time the CMB was emitted".
Perhaps Brian Greene was talking about the size of the observable universe at the time when the CMB photons started traveling towards us. That happened 379,000 years after the big bang at a redshift of 1098 which means the universe was about 84.6 million light years in diameter which, per WolframAlpha, is about half the diameter of the local super cluster of galaxies or about 840 times the diameter of our galaxy.
But it is very important to understand this is all about the observable universe. The whole universe might very well be infinite and might always have been. The big bang did not happen at a certain point in space, it happened everywhere at the same time. The universe did not have an edge, what the photons would reach.
So to conclude: the Big Bang is the zero time limit of the FLRW metric, and it's a time when the spacing between every point in the universe becomes zero and the density goes to infinity. It should be clear that we can't associate the Big Bang with a single spatial point because the distance between all points was zero so the Big Bang happened at all points in space. This is why it's commonly said that the Big Bang happened everywhere. In the discussion above I've several times casually referred to the universe as infinite, but what I really mean is that it can't have an edge.