Frederick Douglass believed that all people are created equal. But he also believed that we weren't just born free: we have to make ourselves into who we are. So education and self-improvement are incredibly important to him. The worst thing about slavery, to his mind, is that it prevents people from improving themselves through education. In fact, he argues that slavery and education are completely opposite things. He works towards making himself free by expanding his horizons though reading. He still has to physically escape, of course, but it's his education that gives him the strength of will to make it happen.
In order to be truly free, Douglass needs an education. He cannot escape until he has learned to read, write, and think for himself about what slavery really is.
Since literacy and education are such an important part of Douglass's growth, the act of writing the Narrative is his final step in becoming free. In a sense, the story he tells in the book doesn't end until he's written the book itself.