Why must animals maintain a stable environment within their bodies?

Homeostasis, my friend.

Homeostasis is the process of maintaining a constant internal environment (temperature, water content, available energy, available oxygen & concentration of wastes in the blood.) Some animals are able to do this and are called endothermic or 'warm-blooded' (like humans). However, not all animals are able to do all of this - and they are referred to as ectothermic or 'cold-blooded.' 

Animals need to be able to maintain a stable internal environment because if they do not, they will most likely die. Animals need warmth to survive, so they need to be able to have a constant body temperature whether it be by using their body to create warmth or having to sit in the sun to keep warm,

Hope this helps.