"Get in the last word" is an idiom. It means to say something that puts an end to the discussion. It might be something that merely attempts to contradict or dismiss what the other person is saying, and in that case there could well be competing attempts to get in the last word. Or it might be something so profound or so obviously true that the other person simply cannot find a suitable reply. I think that is the sense in which Slim used it in your example.
Footnote: gotten is still commonly heard in AmE, but I have been told that it is extinct in BrE.