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Lots of people seem to be asking this. They often seem to get shallow answers that merely assert what is true, instead of drawing or explaining the mechanism. They also seem to not find each other -- if you don't know the answer, it's not obvious that these are all the same question.

Has someone given a high-quality answer that we can refer all these askers to? If not, give it your best shot below.

Here are some related questions. This list is incomplete. Feel free to edit and add to it.

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Interactions make terms significant in regression when they should not be

Why is the intercept in multiple regression changing when including/excluding regressors?

Why could centering independent variables change the main effects with moderation?

Related but not bang on:

What if interaction wipes out my direct effects in regression?

Understanding Simpson's paradox: Andrew Gelman's example with regressing income on sex and height

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