boomerang said:
Firstly what has a black body got to do with microwaves?
I am not too sure what a black body is but neither of ideas as to what one one is
make sense.
Wikipedia's fairly lenghty article on black bodys makes no mention of closed sytems
whatsoever, and even if it did I am not convinced of it's relavance anyway.
All bodies emitted thermal radiation with a spectrum that follows the Black Body form. The peak of this spectrum depends on the temperature of the body. For instance, you are currently emitting black body radiation with in the infra-red, corresponding to your body temperature of around 315 Kelvin. However, your spectrum will not be a perfect black body since you are also reflecting light from other sources, and the clothes and skin of your body will absorb some of that light more that others. Therefore the only way for a body to emit the 'pure' thermal spectrum is for it to not be influenced by any external bodies.
boomerang said:
And how was that calculated?
What temperatures would we have expected and why?
There are two concept here. The first is known as 'decoupling' of baryons and photons and the second is the effects of redshift. These two things together tell us the expected temperature we observe today.
Firstly decoupling. The very early universe was so hot and dense that hydrogen that made up the majority of matter in the universe at the time was completely ionised. That is to say since it was so hot the electrons were not bound around the hydrogen nuclei as in a neutral atom, but the electrons and protons were all moving separately. There was also a lot of photons (light) bouncing around. Now, as the universe expands the density and temperature of this hot soup of ionised hydrogen cooled. We know from experiments the temperature and pressure at which hydrogen becomes ionised and hence when the universe dropped below this temperature the electrons 'recombined' with the protons to form neutral (non ionised) hydrogen gas. This is important because while the gas was ionised, the photons could not travel far before hitting the electrons that were free. Once the universe became neutral photons could travel much more unhindered. This is known as 'decoupling' since this is when the radiation (light) and matter decoupled. What it also means though is that the energy distribution of these photons was as perfect black body spectrum, with the peak corresponding to the known temperature that hydrogen ionises at. Thus we can predict the temperature.
The second point is redshift. As the universe expands, photons released when the universe was smaller have a lower energy when viewed by observers at a late time, when the universe is larger. This is a prediction (that has been confirmed experimentally in labs as well as through observations). As the photons redshift, the form of the black body spectrum remains the same but the implied temperature changes since the peak of the spectrum moves. This modifies the expected measured temperature but in a way that we can predict.
I encourage you to do some further reading on this as there is a great wealth of information out there. A forum such as this can help you with bits you don't understand or have further questions with but to get a good, coherent and well structured explanation of the whole picture it is not the best place.