What does the phrase "on to" mean?
It means, roughly, "proceed to", but has more of a tone of encouragement or excitement rather than a command. You can think of it as having an implicit "Let's move" at the beginning, as in "(Let's move) on to chapter 6!"
This is generally used as an mild enthusiastic exclamation.
It is implied that some subject is moving on to whatever is described. In this case, "Here we go, we are moving on to chapter 6!" is equivalent.