After the Big Bang, the universe was extremely hot - hundreds of millions of degrees. So hot that atoms as we know them could not form. It was also expanding very rapidly, and cooled as it did so. Until about 400,000 years after the start, the high temperatures prevented electrons from settling into orbit around a proton to from a hydrogen atom; they would just get ripped away. We say that matter was ionized, that it was plasma. That hot plasma was opaque to light in the same way the clouds or fog are. We say that it scattered light. This lasted until things cooled down to a balmy 3000K. (0K is absolute-zero. 273K is where water freezes.)
At 3000K, electrons could join protons and form atoms as we know them - no longer ionized, no longer plasma. Once the electrons joined up with protons to make hydrogen that wasn't ionized, these atoms let light pass through pretty easily; the fog had cleared. At the moment when those atoms began to form, which was fairly sudden, light could freely make it's way across the universe, and did. Light began traveling from everywhere to everywhere. (I call it light, but it would not have been in the visible spectrum.)
A satellite called COBE (Cosmic Background Explorer) took the measurements, and they correspond so exactly with the values that the Big Bang Theory predicts, that when overlayed they make a single curve. The difference between them is less than a pixel in that image. This is why XKCD put out a fist-pump for science, but it isn't the part that gives me chills.
That plasma cloud is called the Last Scattering Surface because it is the last thing that ever touched (scattered) one of these photons before they reach us. Photons bounced off the Last Scattering Surface, a cloud of hot plasma, 13.72 billion years ago, traveled across the universe for essentially as long as it's been around, and have arrived here on Earth. They are coming equally from all directions, these ancient travelers, and we are bathed in them constantly. Most of the radiation is up at 160Ghz, but some of it also exists down at radio frequencies, and this Cosmic Microwave Background makes up about 1% of the white radio noise we have here on Earth.
This means that when you see snow on an old television, or hear static on an FM radio, a little whisper in that is the recombination epoch. The universe is reaching out to us from a time so ancient that stars had not yet been born. Those empty TV channels and radio stations that we thought of as nothing, they were actually playing the beginning of everything.
Shhhhh, I'm listening to the Big Bang.