How should I understand the phrase in bold?
Jesse remained on dry land only long enough to document his voyage in a book called Lionheart: A Journey of the Human Spirit, and to plan his next adventure. He is now off on what has been named “The Journey of Kijana,” a two year around the world excursion crewed by five young people.
I don't understand what it means to be "off a book/story".
"To be off" can mean to start a trip or journey, or move away from somewhere. Jesse wrote a book called "Lionheart: A Journey of the Human Spirit", and planned his next adventure (another journey); he is off (has started) on that journey, which has been named "The Journey of Kijana". He will probably write a book about it, using that as a title, when he returns.
3 Starting a journey or race; leaving.
‘we're off on holiday tomorrow’
to be off on [a trip, journey,etc.] means:to have left a place and begun the trip, journey,etc.
In fact, there is an idiom we use a lot with this: Shall we be off? or Let's be off. Meaning: Let's leave now to go somewhere.
