Well, it doesn't say you have to win quickly.

I was skeptical at first, but consider it this way: At each step you make a subtree simpler, and then insert an arbitrary number of copies of the simpler subtree. Eventually you must end up with a large number of copies of the simplest possible subtree, a single node off the root. Those don't grow the hydra when removed, so you you chop them all off and then win.

I found I could see this intuitively if I chopped the top-most head of the most-complex tree for the first several rounds, in most configurations; you'll see whatever tree you're working on get wider, but shorter. It helps to lower the starting number of nodes to 7 or so, as well.