It means don’t accept the boundaries and limitations that others set for you. In the story, this statement is written on an epigram as a warning to others. In the context of the story, it means: “don’t allow social control to destroy your creativity, imagination and thirst for knowledge.” More generally, the statement applies to any set of standards or any external authority that would limit your freedom. The lines on the paper represent the rules and restrictions enforced by society or government. Writing “the other way,” or across the lines (rather than within them) represents and act of rebellion. It means sometimes you need to break the rules in order to express your individuality and autonomy.
It means to think for yourself and to not follow the crowd. Write where and how you want to, you don't have to follow all the laws and rules.