I think the reason people believe they do good is because deep down inside, they believe that there is an objective good, that some actions, beliefs and morals are true regardless if everyone believes in them or not.
Otherwise, if moral values are all subjective, then no one's moral values are any better than another's, and therefore people shouldn't be mad at one another or try to change each other for having different moral values, no matter how 'extreme' or 'radical' they are in regards to societal norms. Any such act of putting a set of subjective morals above another set of subjective morals implies that one subject is greater or more valuable than another subject.