The big bang theory has another explanation for the CMB. Within a tiny fraction of a second after the big bang, the universe was about the size of a basketball and was expanding trillions of billions of times faster than the speed of light today. Matter was so hot that it was a plasma (electrons and nuclei could not combine). Therefore, light, scattered back and forth by all the free electrons, became black-body radiation.
Approximately 400,000 years after the big bang, the universe, still expanding, had cooled enough for electrons and nuclei to combine into atoms. Without interference from free electrons, light in the form of the cosmic microwave background radiation could now be seen.