There was a time when religious people believed that human urges that are not in line with popular morality were "whispers" or suggestions from the demons (evil spirits). So, my interpretation of "fight one's demons" is to resist temptation. HOWEVER, I believe the correct (or at least more commonly used) expression is "FACE one's demons," not fight. Which means to confront one's fears. I can imagine how one would confuse these two expressions