The phrase "fight the system" means to try to go against the way things are normally done in some institution, like a school or a workplace (or even a whole society).  So, in this case, Jerry is fighting the system.  He is not going to do what the people with the power tell him to do.

The reason that most people would say it is important to do this is because the system is not always right.  If you just go along with what everyone else does, or with what you are told to do, you sort of risk losing your conscience.  You risk getting to the point where you are not your own person -- you are just someone who does what they are told, no matter what.