Why does strenuous exercise stimulate deep breathing that continues after the end of the exercise?

Answer 1

its called your "oxygen debt" when you excersise your bodies level of CO2 increases dramatically, which stimulates the need for Oxygen, and the rapid breathing. after excersise your levels of CO2 are still high so your body continues to keep your respiratory rate high to normalize the amound of CO2 in your blood stream

Answer 2

Much of the lactic acid is being converted back into glucose in the liver and ADP is being converted back to ATP (phosphorylation). Because both of these reactions demand energy input, the body has to supply energy. Much of this energy is being supplied by AEROBIC lipolysis (hence the heavy breathing). This also explains why (contrary to popular belief) weight training burns a lot of fat (believe it or not). Here is a free website that explains it in more detail.