To extenuate is to make less of something or try to minimize its importance. The fact that you walked your little sister to school because she missed the bus might extenuate your teacher's response when you show up late.
Extenuate goes back to the Latin verb extenuāre, meaning "to thin" or "make thin." Someone might have a strong case against you for doing something wrong, but an added consideration can make less serious, or extenuate, the circumstances that led to it. If you give an excuse that doesn't really relate to what was done, it won't extenuate the punishment. If something has a real bearing on what went wrong, it will extenuate the response, and you'll get in less trouble.