Why can we see the cosmic microwave background (CMB)?

The cosmic microwave background does not originate with the big bang itself. It originates roughly 380,000 years after the big bang, when the temperature dropped far enough to allow electrons and protons to form atoms. When it was released, the cosmic microwave background wasn't microwave at all- the photons had higher energies. Since that time, they have been redshifted due to the expansion of the universe, and are presently in the microwave band.

The universe is opaque from 380,000 years and earlier. The galaxies that we can see only formed after that time. Before that, all that is observable is the CMB.

The CMB we see is the state of the universe when it became transparent, 380000 years after the Big Bang. Our lines of sight away from us can't see any farther than this. Think of the Sun, which is a big ball of gas, but appears to have a surface because that is where the accumulated gas along the line of sight to the center has become opaque.

The universe became transparent after 380000 years because the cooling caused by its expansion allowed ionized gas to recombine into neutral hydrogen and helium, which is transparent to visible light. At this time, gravity could begin to work on the slight density variations causing the denser parts to slowly become more and more dense until nuclear reactions would start at the densest core, marking the beginning of star formation. But because the universe was still hot and the initial density variations very slight, it still took millions of years for the first stars to form.

While in the neutral state, the universe was transparent to visible light but opaque in the far ultraviolet due to absorption from the electrons around the neutral hydrogen atoms. When we try to see back to this time with Hubble and other optical telescopes, the very high redshift has moved the far ultraviolet through the visible spectrum and into the infrared, making most of the neutral period invisible to us. The James Webb Space Telescope (JWST), will have detectors that work far enough into the infrared that we can see what happens during the neutral time. After hundreds of millions of years, the ultraviolet light from hot stars reionized the universe making it once more transparent to the ultraviolet (and now transparent in the visible as well owing to the low density after hundreds of millions years of expansion).

The idea that we see the actual moments of the Big Bang when we look at the CMB arises because the density variations could not change during the first 380000 years, so those we see in the CMB were there from the first instants of the Big Bang and represent quantum-mechanical fluctuations present at the very start.

Dear Carson: To understand why we see today the CMB radiation, after 13.7 billion years after the BB we do not need to know GR at all. Let's use a story to simply illustrate the problem, and get the solution. Suppose you awake in the middle of your street and you find yourself sorrounded by a terrible heavy and dense fog, with a yellow glow all over. So heavy that you can not see the floor. Even your hands are invisible. We say this fog is "opaque", and has a glow. This special fog (Of this story) stretches to infinite. All of a sudden, as an divine order, the heavy fog becomes transparent. Every spot of the space where the fog is, becomes transparent. What will you see? Are you going to see transparency all over at once? NO WAY! Normally, you can see your sorroundings thanks to light. But light takes time to reach your eyes! So, in spite that every spot gets transparency at once, you won't see that! First you will see yourself, then the street and houses of the neighborhood backed by the brightness of the fog in rapid retreat. One second and a half later you will see the moon, but the whole sky will be shining with a yellow glow. In eight minutes more you can see the sun. And in 4.3 years the closest star (Alpha Centaurii). But the yellow glow will stay there all the time on. Forever! You are in the center of a hollow sphere whose radius is growing at the light speed! What you see is what is called "the surface of the last scattering" The cosmologic expansion of the space was the cause for the reduction and redshift of the originally "yellow glow". Actually the original fog was incandescent gas, same as the interior of the sun.
I hope this can help to understand why we can see yet the first light of the universe!