Why does stale bread go soft in the microwave?
I believe this has to do with its moisture content. While certainly dry, it still has some moisture, and what little moisture is left gets excited by the microwaves, heats up (the heat helps it soften too) and steams the bread essentially. I think that MaxW's explanation of glutens for the question 'Where does the moisture come from?' is a likely explanation. 

Note that if the bread is sufficiently dry enough, this little microwave trick wont help. It is usually advised that you add a few drops of water to the bread before heating up in the microwave in this way. This is to ensure the bread doesn't actually become harder in the microwave, as it boils the water and will drive it off if microwaved in excess. 

Also, note that bread WILL become blackened, charred and start smoking if left in the microwave long enough. I did this by typing 30 minutes instead of 30 seconds. It started smoking around 5 minutes.