"goes against every fiber of my being" means whatever you are asked to do contradicts or goes against everything you stand for or believe in. For example, if I was a passionate vegan and you were trying to pressure me to eat a hamburger, I might say, "No! It goes against every fiber of my being!" It's basically a way of saying no with much more emphasis. Note that you would never say this unless you felt passionately about the issue. For example, if I just don't LIKE hamburgers and you are pressuring me to eat one, I would just say "no."

It means that a certain idea is the contrary of you believe in. A simple example of it would be if you are asked to do something that stands strongly against what you see as right such killing a person.