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.