Why does CMB radiation propagate towards us? There is something with CMB radiation that does not sit well with me... It seems very counterintuitive that we are able to see it. If CMB radiation formed at the early phases of the universe, would it not make sense that it expands and propagates "outwards" with the Big bang so that we would never see it? Most space is out of our reach since it does not belong to the observable universe. But CMB radiation formed earlier than most space, should it not also be far outside the observational universe on its way away from us?
How should I visualize the CMB radiation? I have a master in theoretical physics but I never went back to understand this.
Any help in understanding this is very welcomed.
 A: Imagine 13 billion years ago, the Universe is a hot soup with many photons bouncing around in random directions. Every direction you look, hot soup photons.
Now expand everything and come to today. The same hot photon soup is still here, but redshifted. Every direction you look, wherever you are, soup photons. We are in the same universal cloud of photons.
The CMB is essentially our view from inside a photon gas or perfect blackbody.
The issue with understanding the "observable" Universe is that there are many horizons to be confused between: https://en.wikipedia.org/wiki/Cosmological_horizon
Since the Universe is expanding, distance travelled by a photon 13 billion years ago is worth more distance today (similar to how money a long time ago was worth more). This means that we can see photons from a very long distance that were emitted in the past (particle horizon), even though we can never visit the place where it came from if we started today (event horizon).
A: The biggest misconception I see in the question is the idea of the Big Bang as something that propagates "outwards", like an explosion. There is no outwards direction, the universe didn't expand into something.
Even though recent observations seem to suggest that the universe is closed, for the sake of simplicity let me assume that the universe is flat and infinite.
The first thing to notice is that if the universe is infinite now, it was infinite also at shortly after the Big Bang, it was just more dense. Imagine a flat plane on which it is drawn a lattice of dots, equally spaced.

The plane is the universe at a certain time. The dots represent objects in the universe, electrons, atoms, stars, whatever.
Now, imagine scaling up the plane, making it bigger. Of course the plane is infinite, so scaling it doesn't change its size, but the dots get further away from each other while keeping their size fixed (otherwise, you wouldn't be able to tell that an expansion took place).

You see that the dots aren't expanding into anything, the universe (the plane) was already infinite, it didn't grow in size.
Critically, there is not a center of the expansion. The distance from any two dots has doubled (in this example), regardless of their position.
Now, let's talk about the CMB. Imagine that at time $t_0$, every dot emitted a pulse, an expanding circular wave. This wave symbolizes the photons of the cmb that are emitted at the same time from every point towards every direction.

The last pictures refers to our situation. Earth is the black dot that is touched by the wave fronts. We see the CMB coming from every direction (in the picture only from four directions) because it was emitted from everywhere.

A: The CMBR did not occur in a single location, but rather throughout the entire early universe (from over 13 billion years ago).  Therefore, any theoretical observer throughout the full history of the universe is able to see the CMBR passing through their area of the universe in their present moment (however, with a different thermal signature).  What we see today is light that has traveled through a distance of over 13 billion light years.  This total distance is comprised of both the initial distance (what your distance would have been from a certain region of the universe at the time the CMBR was emitted) plus the additional distance created by the stretching of spacetime (and hence the observed cooling of the CMBR).
Since the CMBR is not an event (a point in spacetime defined specifically by three spatial dimensions and one temporal) but rather occurred everywhere, you simply need to align the distance traveled with the time that has transpired and you will be able to observe the CMBR that was emitted from the corresponding region of the universe at the time the CMBR was emitted.
A: I struggled with visualising this for some time. What we are seeing as the CMB is the "surface of last scattering"; in other words the (receding) surface where photons were emitted after recombination.
I found Lineweaver's "surface of last screaming" a helpful analogy:

Consider an infinite field full of people screaming. The circles are their heads. You are screaming too. (Your head is the black dot.) Now suppose everyone stops screaming at the same time. What will you hear? Sound travels at 330 m/s. One second after everyone stops screaming you will be able to hear the screams from a `surface of last screaming' 330 meters away from you in all directions. After 3 seconds the faint screaming will be coming from 1 km away...etc.

https://ned.ipac.caltech.edu/level5/March03/Lineweaver/Lineweaver7_2.html
