I think it means he became excited and enthusiastic about writing the book on Lincoln."ON fire" here means being very excited and wanting to start work,like a burning in his heart...

It never hurts to give the full context. From Google I learn that this is from Dale Carnegie's book "Lincoln the Unknown," and a longer passage is "i went over the British Museum library and read a number of Lincoln books; and the more I read, the more fascinated I became. Finally I caught on fire and I was determined to write a book about Lincoln myself." He is using "heat" as a metaphor for "interest." He got more and more interested--hotter and hotter--until he "caught on fire."