The recirculating button opens & closes a vent that allows fresh air from outside to enter the cabin. When the switch is pressed, and the little light is on, the vent is closed and no outside air can get in - the air inside the cabin is recirculated.
You might want to to this when there are fumes or bad smells outside, say when driving in heavy city traffic or past a farmers field when he's just spread it with manure. Pressing the button means these bad smells won't enter the car.
But the downside is that with no fresh air coming in, the moisture in your breath gradually builds up and the windows steam up. Even with the blowers on full you're just moving that moist air around, so the windows never really clear.
Opening the recirculating vent by pressing the button again means fresh air comes in and your windows will clear a lot faster.
Re: the air conditioning - its for the same reason. Air con also acts like a dehumidifier, removing moisture from the air. This means the windows will clear faster, so when you put the right-hand dial to 12 o'clock (all air directed at the windows) the car is programmed to switch the air con on, to help you demist your windscreen.