What the hell is in that click?

I'm not seeing that there's anything so mysterious here. From your description, to click is to realize an implication of your beliefs so quickly that you aren't conscious of the process of inference as it happens. You add that this inference should be one that most people fail to draw, even if the reasoning is presented to them explicitly.

I expect that, for this to happen, the relevant beliefs must happen to be

1. cached in a rapidly-accessible part of your mind,

2. stored in a form such that the conclusion is a very short inferential step beyond them, and

3. free of any obstructing beliefs.

By an obstructing belief, I don't mean a belief contradicting the other beliefs. I mean a belief that lowers you estimate of the conditional probability of the conclusion that you would otherwise have reached.

When you are trying to induce other people to click, you can do something about (1) and (2) above. You can format the relevant beliefs in the most transparent way possible, and you can use emphasis and repetition to get the beliefs cached.

But if your interlocutors still fail to click, it's probably because (3) didn't happen. That is, it's probably just a special case of the usual reason why people fail to be convinced by an argument, even when they grant the premises. People fail to be convinced because they have other beliefs, which, when taken into account, seem to lower the overall probability of your conclusion. So, typically, a failure to click is no more mysterious than a general failure to be convinced by arguments.

On a more cynical note, I'm pretty sure that the "click" is almost the only decision procedure for the vast majority of people*. When a question arises, one answer will seem to be manifestly the right answer, and the rest will seem obviously wrong. When they change their mind, it will be because another answer abruptly seems to be manifestly the right answer. If no answer clicks for them, they will just chalk the problem up as "mysterious".

*Here I'm using "click" to include inferences that aren't necessarily rare, and which might in fact be very common.