cosmology - How is it possible that the CMB approaches the earth from all directions?

The Big Bang happened everywhere and the recombination of electrons with nuclei - the thing that causes the CMB - also happened everywhere. So in every direction you look you can see the radiation from that recombination.

But that recombination also happened a long time ago and so, given light has a constant velocity, the places where we can see the recombination take place are also a long way away. And, as a result of the expansion of space-time, that radiation is very red-shifted, so instead of the recombination looking like it is happening at roughly 4000 degrees Kelvin, it looks like something happening at 3 degrees Kelvin.

That expansion is also what makes it look like the distant galaxies are all moving away from us and the further away the faster they seem to be moving.