"Someone please explain to me in what sense that "joke" was supposed to be funny?"
It was poking fun at the fact that one has to enter rehab these days if they dare to utter that word. That was the point and unfortunately we've gotten so used to letting the left tell us what is and is not appropriate that even conservatives toed the line and condemned her lousy joke.
That's a perfectly valid thing to joke about, of course, but: what on earth does it have to do with John Edwards? Why didn't she make a joke about, say, airline food and use the phrase "John Edwards" in the same sentence?
Because that. would. make. no. sense. Right?
Well, neither did this "joke". It's a total non sequitur.