It's a phrase commonly meant to say that you can easily see a person's motives and that you can easily understand everything about a person.
But if the book we're referring to is even a mediocre book at best, that's not what this statement means in the least. Any book with any level of greatness to it reveals its secrets slowly.
The reader, upon opening the book and reading the first page, invests herself in the book's ride. The reader decides to stick with the book through thick and thin as all of its secrets are slowly uncovered layer by later.
Some revelations will be joyous, some reassuring, but others still will be awful, shocking things. The reader, however, sticks with the book through the thick and thin, both eagerly and patiently peeling back the layers.
So, really, the phrase "I can read you like a book," is saying, "I'm willing to read you like a book. I'm ready to embark on this journey with you and get to know you gradually, and I'll stick with you through everything."