They increase cache locality and runtime performance of iteration and equality checking. If that answer isn't sufficient, try watching the ~30min Clojure West video on the github page. The speaker seems to be a bit new to public speaking but his talk on the subject was easy to follow and informative for me.
The talk at Clojure West wasn't given by myself, but I found it worthwhile linking. Rather it was given by someone who independently picked up my research results and replicated them in the context of ClojureScript (https://github.com/bendyworks/lean-map). The authors independently confirmed the performance improvements of CHAMP over HAMT that I was observing.
Why should that be the case? Can you point to sources where you read that? In my experience, CHAMP clearly has advantages over HAMT in this scenario.
To answer your questions about using vectors of coordinates of keys: it depends on the design implementation of the vector's hash code, regardless if you use HAMT or CHAMP.
Using collections as keys in other collections is in general a performance sensitive subject. The available HAMT implementations in Clojure and Scala fail to deliver here.
The case study in Chapter 3.7 nests hash-sets into hash-sets (i.e., Set.Immutable<Set.Immutable<K>> sets). The CHAMP implementation yields minimal speedups of ~10x over Clojure and Scala due to the way it calculates and incrementally updates the collection's hash code.
I was there and you are right, he was very nervous. However his talk was excellent and approachable for everyone reading the above comment. Explains the differences quite well. Certainly worth a talk. I've asked on the clojure slack group if the proposal had gone anywhere and it seems to have stalled unfortunately.