He breaks me up = he makes me laugh. Don't break me up is not idiomatic: don't make me laugh is much better. It also carries the idea that whatever the topic of discussion is is nonsense.
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Mr WordyI think this expression is specific to AmE, if it makes a difference. As a BrE speaker, I'm unfamiliar with it (though with a reasonably helpful context I'd guess its meaning). The similar expression that I know is "crack somone up".
Yes, we use 'crack' in the same context.