Due to scope neglect, framing effects, and other cognitive biases, the result of an expected utility calculation executed correctly may produce an answer different from first intuition, making it "intuitively unappealing". If you can tell that it's probably the intuitions that went wrong and not the calculation, the skill shut up and multiply is the ability to accept that, yes, sometimes the expected utility math is correct and we need to deal with that. Contrast do the math, then go with your gut. If you're not sure which of these applies, use "do the math, then go what your gut" until you've built up more experience.
The specific application of Shut Up and Multiply to the Torture versus Dust Specs case has proven quite contentious. One reason this case was cited as an exemplar of where "shut up and multiply" should apply was a claim that the usual reasoning behind answering "SPECKS" can be reduced to circular preferences.