"Can anyone help me to explain why early rise will invoke the heel turn? Thanks"

Especially as there are heel turns which do not formally feature "rise" at all, it will probably be more practically useful to focus instead on the difference in use of the knee during the first step.

In a follower's heel turn, the knee (of both partners) remains "braced" which is to say relatively straight in comparison to in a conventional (waltz-like) turn where there is a notable bending of the knee to absorb arrival, and a re-straightening during departure. Simply using his own knee in one way or the other will be sufficient to allow the man to communicate his intent to a follower who is moving with him.