To understand that sentence the context is important. “would allow people to change the narrative by telling their own stories." In a political or news context I believe the above sentence has the following meaning: The "narrative" I believe they are referring to is the current status quo/commonly held beliefs about something. People "telling their own stories" helps provide more information about a story that helps other people better understand their point-of-view. For example, a high school student talking about their experience during a school shooting may help people understand that two greatly underfunded areas in school budgets are mental health counselors and having enough support staff to make sure no child commits acts of horror. The student's stories will put extra emphasis on addressing the need for a greater focus on the mental health narrative.

It suggests that the person who wrote this allows people to make changes in the storyline/plot of his/her novel or short story by using their own fantasies or experiences.