Organisms have evolved physiologically and anatomically to adapt. How have their systems evolved to maximize their purpose of maintaining homeostasis and equilibrium? Explain.

I'm unsure whether "how have their systems evolved" is asking for a likely method of evolution or is asking about what specific things an organism does to maintain homeostasis and equilibrium.  If it is the second option, the question is also confusing because it doesn't mention which organism is being examined.  Not all creatures are capable of maintaining equilibrium and homeostasis the way that humans/mammals do.  
If the question is asking about a method of evolution, then Darwin's explanation using adaptation and fitness levels is probably the most appropriate. Because of sexual reproduction, new genetic combinations are always occurring within any given species.  Those changes in the genetic code will cause an organism to be better or worse adapted to a particular environment.  An organism that is better adapted is said to be more "fit."  Fit organisms tend to survive (survival of the fittest).  Because they are surviving, they are more likely to pass on their particular adaptation. Generations will go by and eventually the entire species will have that adaptation.  In other other words, nature selects which adaptations a species will evolve with.  Applied to the original question, that means that at some point in history, an organism developed some kind of adaptation that allowed him/her/it to more effectively maintain homeostasis.  That in turn somehow allowed that creature to better survive in that particular environment.  Its progeny then inherited that adaptation until that adaptation eventually spread through the entire population. 
https://www.ndsu.edu/pubweb/~mcclean/plsc431/popgen/popgen5.htm