The intuition (which is described in the original article, by the way) is that in this puzzle, whoever actually finds the solution can take the entire reward for themselves.
We've expanded this idea into a rigorous research paper that's currently undergoing peer review, but we may release a preprint soon.
This destroys the ability of pool members to trust that their nonces, where valid, will actually be used to claim a reward for the pool.
I'm not familiar with the work, but I assume there are validated zero knowledge proof protocols that prove you know a partial hash inversion of a particular quality, like the Pinocchio one referred to.
Edit: with that said, I don't agree that eliminating mining pools is a good thing; that just rewards those who can enforce cooperation by some other means (say, having the capital to personally run the farm), which has a far more unequal distribution than mining pools.
In order for Bitcoin to work, in the sense that you don't have to trust anyone for it to be fair, then no group can ever have 51% of the hashing power of the network. It has now been proven that a mining pool can reach that threshold. It has NOT been proven that it is practically possible to buy that much hashing power. We should get rid of mining pools because they are the current cause of this crisis. We can cross the other bridge when we come to it (which it seems unlikely we ever will).