Question

How is the cosmic microwave background a logical conclusion from the idea of the Big Bang?

Answer 1

The idea of big bang was motivated by the observation that the universe is expanding i.e. distance between any two galaxies is increasing with time. So, extrapolating back in the past , all the matter and radiation in the universe today can be traced back to a tiny region of space in the far past.

The essential point in the idea of big bang is that the universe, at least that part of the universe from which we have received radiation so far, is an outcome of expansion, by a huge factor, of what was once a tiny region of space. This tiny region of space was causally connected i.e. different parts of the universe we see today have either communicated with each other or have a common source in the past. If thermal equilibrium was achieved at some point in the far past, in this tiny region of space that has expanded to become all of the accessible part of the universe today, the radiation produced in that thermal equilibrium leads to cosmic microwave background we see today although we do not have thermal equilibrium today in all of the observable universe.