This exercise makes a whole lot of sense. It's using the eccentric contraction phase to treat tendonitis. This really excellent NY Times article describes the rationale as applied to tennis elbow, which I posted before on epicski in response to a tennis elbow thread. I have used a modification of the exercise to treat my own tendonitis on the opposite side of the elbow, "golfer's elbow". That it should work for Achilles tendonitis makes perfect sense. Actually- using neonorchid's description above, continue to use both legs to start the movement even when things get better. The idea is to pre-load the symptomatic side by contracting the calf muscles under minimal stress, then maximizing the stress in the negative phase of the movement.
https://well.blogs.nytimes.com/2009/08/25/phys-ed-an-easy-fix-for-tennis-elbow/
Maybe some coming to this thread who happen to have tennis elbow can benefit as well.
Edit: The article used to link to a YouTube video of the movement. That apparently has not survived the intervening years. Too bad. If a picture is worth a thousand words, that video was worth ten thousand.
If we're now talking about tennis elbow, my favorite treatment is self massage with a lacrosse ball. First discovered this with a light case, and was cured in a couple three weeks. Now, will use after a rough session before symptoms appear. I play weekly and symptoms haven't reappeared in over a year. I'm now playing with heavier strokes and equipment.
The twisting of the rubber stick didn't do much for me and others I know.