It may seem that when bread goes stale is a result of it losing water and getting more and more dry. If that were the case, it would make sense to think that putting the bread in a humid and cold environment will slow down water evaporation and therefore the staling process. This would indicate that putting bread in the fridge, we should be able to keep it fresh longer.
I conducted the following experiment. I cooked pre-baked bread in the oven, sliced 4 pieces of it and store them in 4 different places: in open air, wrapped in a cloth at room temperature, in the fridge and in the freezer. After 10 hours, which one do you think was the hardest? In the next picture I have arranged them from the hardest to the softest:
The hardest one was of course the one that was in the freezer but then, surprisingly, the next hardest one was the one that was in the fridge! The one that had been left in open air was also hard, although less than the other two and the one I had kept inside a cloth had acquired a chewy consistency.
Actually in the process of going stale bread does not lose water, but rather the opposite. What happens is a phase transition. The starch molecules that constitute the bread slowly crystallize to a more rigid form, making the bread harder and giving the impression that it is drying out. As it turns out, to crystallize they starch molecules need to associate with lots of water so what they are doing is taking free water molecules from the bread and the surrounding air and trapping them in a crystal form.
This also explains why the piece of bread that I kept in a cloth did not get harder but rather chewy (the effect is bigger if you keep it in a plastic bag). In that case, the starch can not access as many water molecules in the air and therefore does not crystallize.
One last thing to notice is that this is also the reason why cakes stale at a much slower rate that bread. This is because cakes have sugar and sugar loves water, so the sugar absorbs some of the water around which are then not available for the bread to absorb and then crystallize slowing down the staling process.
The speed at which this crystallization takes place depends, among other things, on temperature and it has its peak at around 4ºC (39F) which is why it stales quicker in the fridge than it does in open air.
The good news is that this process is partially reversible. All you need to do is heat the bread a little bit and it will look like fresh for some minutes (when it cools down again it will be worse than before because in the process it will have lost moisture). I put my 4 slices of bread in a preheated oven with a very low temperature 80ºC (176F) for only 3 minutes and it was good as new :).