This is sung from the point of view of a guy who's girlfriend (or best friend, maybe) is under the control of a chemical addiction. In Alcoholics Anonymous, alcoholism is referred to as a "Monkey on your back", because it's a big, ugly, heavy, and sometimes obvious problem that you have to carry around with you no matter where you go. In this case, the "Monkey" has such a tight grasp on the person that he feels that it is in her soul.
This is sung from the point of view of a guy who's girlfriend (or best friend, maybe) is under the control of a chemical addiction. In Alcoholics Anonymous, alcoholism is referred to as a "Monkey on your back", because it's a big, ugly, heavy, and sometimes obvious problem that you have to carry around with you no matter where you go. In this case, the "Monkey" has such a tight grasp on the person that he feels that it is in her soul.