Where does the microwave background radiation come from?

The microwave background radiation is left over from the big bang. Originally the whole universe was so hot it was a plasma and the electrons had so much energy they were wandering about separate from the nuclei. About 300 years after the big bang the universe cooled enough for the electrons to get trapped by nuclei-forming atoms. This released an immense amount of energy in the form of light. Originally this would have been in the form of X-rays or ultra violet with a wavelength of less than a thousand trillionth of a metre. As the universe expanded, the light was stretched too, to the couple of millimetres it is on average now. Microwave background radiation is still around now because the universe is very empty and it hasn't all hit anything solid yet.