Related to: Can Counterfactuals Be True?, Newcomb's Problem and Regret of Rationality.
Imagine that one day, Omega comes to you and says that it has just tossed a fair coin, and given that the coin came up tails, it decided to ask you to give it $100. Whatever you do in this situation, nothing else will happen differently in reality as a result. Naturally you don't want to give up your $100. But see, Omega tells you that if the coin came up heads instead of tails, it'd give you $10000, but only if you'd agree to give it $100 if the coin came up tails.
Omega can predict your decision in case it asked you to give it $100, even if that hasn't actually happened, it can compute the counterfactual truth. Omega is also known to be absolutely honest and trustworthy, no word-twisting, so the facts are really as it says, it really tossed a coin and really would've given you $10000.
From your current position, it seems absurd to give up your $100. Nothing good happens if you do that, the coin has already landed tails up, you'll never see the counterfactual $10000. But look at this situation from your point of view before Omega tossed the coin. There, you have two possible branches ahead of you, of equal probability. On one branch, you are asked to part with $100, and on the other branch, you are conditionally given $10000. If you decide to keep $100, the expected gain from this decision is $0: there is no exchange of money, you don't give Omega anything on the first branch, and as a result Omega doesn't give you anything on the second branch. If you decide to give $100 on the first branch, then Omega gives you $10000 on the second branch, so the expected gain from this decision is
-$100 * 0.5 + $10000 * 0.5 = $4950
So, this straightforward calculation tells that you ought to give up your $100. It looks like a good idea before the coin toss, but it starts to look like a bad idea after the coin came up tails. Had you known about the deal in advance, one possible course of action would be to set up a precommitment. You contract a third party, agreeing that you'll lose $1000 if you don't give $100 to Omega, in case it asks for that. In this case, you leave yourself no other choice.
But in this game, explicit precommitment is not an option: you didn't know about Omega's little game until the coin was already tossed and the outcome of the toss was given to you. The only thing that stands between Omega and your 100$ is your ritual of cognition. And so I ask you all: is the decision to give up $100 when you have no real benefit from it, only counterfactual benefit, an example of winning?
P.S. Let's assume that the coin is deterministic, that in the overwhelming measure of the MWI worlds it gives the same outcome. You don't care about a fraction that sees a different result, in all reality the result is that Omega won't even consider giving you $10000, it only asks for your $100. Also, the deal is unique, you won't see Omega ever again.
Yes, that is the problem in question!
If you want the payoff, you have to be the kind of person who will pay the counterfactual mugger, even once you no longer can benefit from doing so. Is that a reasonable feature for a decision theory to have? It's not clear that it is; it seems strange to pay out, even though the expected value of becoming that kind of person is clearly positive before you see the coin. That's what the counterfactual mugging is about.
If you're asking "why care" rhetorically, and you believe the answer is "you shouldn't be that kind of person", then your decision theory prefers lower expected values, which is also pathological. How do you resolve that tension? This is, once again, literally the entire problem.
Well, as previously stated, my view is that the scenario as stated (single-shot with no precommitment) is not the most helpful hypothetical for designing a decision theory. An iterated version would actually be more relevant, since we want to design an AI that can make more than one decision. And in the iterated version, the tension is largely resolved, because there is a clear motivation to stick with the decision: we still hope for the next coin to come down heads.