Learn Dutch: Neem mij niet kwalijk dat ik het zeg. - Do not take it amiss that I say it.

Even though it doesn't use the exact structure Niet dat..., the phrase Neem mij niet kwalijk dat... serves a very similar discourse function. It is a pre-emptive apology used right before you say something potentially critical or offensive, softening the blow before the listener can get defensive.

I usually hear neem me niet kwalijk used just to say "excuse me". Is the meaning different here?

It is the same core expression, but used in a more literal way. On its own, it is a polite "Excuse me" (like when bumping into someone). But because kwalijk nemen literally means "to resent" or "to take ill", adding a dat-clause allows you to specify exactly what the listener shouldn't resent you for doing.

Why is the phrase split up into Neem mij niet kwalijk?

The base expression is iemand iets kwalijk nemen (to take something amiss from someone), which functions like a separable verb. Because this is an imperative (a command), the conjugated verb Neem goes at the start, and the separable part kwalijk goes to the end of that main clause, wrapping around the object mij and the negation niet.

Could I translate "for saying it" directly as voor het zeggen?