Until the Universe was 380,000 years old, it was filled with a gas of protons an electrons. There was also radiation, in thermal equilibrium with the matter, and because it was so hot, the protons and electrons couldn't form neutral hydrogen, since every time it "tried", an energetic photon would knock off the electron.
This gas was everywhere. And photons traveled and scattered in all directions:
Photons (purple) scatter on free electrons (green), and both are mixed with protons (red).
380,000 years after the Big Bang, the temperature had fallen sufficiently that neutral atoms could form (this is called recombination). The radiation, which until now had scattered continously on free electrons, could now stream freely between the atoms (this is called decoupling).
So they did. Still in all directions:
This free streaming is still taking place. Photons travel in all directions, and are everywhere. The photons that you are able to see, are the ones that started out at a particular distance from you, and in a particular direction, but other photons started out at smaller and larger distances, and in other directions. You just don't see them, because you happen to be right here. But a person in another place of the Universe would see the same as you.
The photons that we observe as the CMB come from a region we call the surface of last scattering, because it corresponds to the surface of a shell centered at us. But there is nothing special about this "surface", except that is consists of all points in the Universe that are so far from us, that it takes a photon roughly 13.8 billion years to travel. And because of the expansion, these points are now roughly 47 billion lightyears away from us.
In the figure below, the arrows show CMB photons. All have the same length; they start where they were emitted and end where they are today. What we observe as the CMB are all arrows that end at the Milky Way (in the center). Other arrows may be observed by other observers in other galaxies that have their own surface of last scattering around them.