I've been using quaternions for 15 years now and I still don't have a good mental image of why they work. I've just learned to accept the operations and what they do.
Visualising Quaternions is a book which goes into much beautiful depth about what they represent and how they behave, if you fancy digging deeper than the everyday practicality of them.
I had never heard of quaternions so I Googled the book title you suggested. In addition to the Amazon link [1], I also found a question on the gamedev SE where they talk about building a mental model of quaternions [2], which I found informative.