I love the accessibility of presenting this very complex philosophical thought experiment in the form of fiction. He walks you through some of the analysis you need to be making in the beginning,
with the narrator-philosopher basically teaching you, but as the story goes on, you are increasingly left to your own devices to unpack the consequences of the increasingly bizarre scenario. By the end, it's completely stopped being a lecture and is just an entertaining story, but still no less instructive because of the way it captures your imagination and makes you want to work through the ideas.
Agreed, Dennett hit all of the high points of the who-am-I philosophy in a really entertaining story. It also reminded me of a great Greg Egan story, Learning To Be Me: