I was reading the wiki page about paraprosdokians and I don't understand why the phrase:
Take my wife – please!
is classified as one.
Paraprosdokians are basically sentences and phrases with surprising endings that are so unexpected that it changes the expecting meaning of the early phrase. It basically plays on the double meaning of a particular word. In this case, it is the word take.
When take is used in the expression "Take my wife...", it is usually used as an instance or example in support of an argument: "let’s take Napoleon, for instance". Ending the phrase with "- please!", which is used to add urgency and emotion to a request, would change the expecting meaning of the early phrase, "take my wife", and therefore turning the entire expression into a paraprosdokian.
The joke is certainly an example of a parawhatsitcalled. It is—as the OP's link notes—a famous one-liner by Henny Youngman. From his wiki:
So, the OP isn't the only one who "didn't get it". The joke's author didn't either.
Then again, I suppose that it isn't as funny in writing as it is when a professional comedian delivers it (probably due to the fact that the strategic pause and inflexion are lost when written). Furthermore, when a comic says it, you are in many ways searching for and expecting the pay-off. Youngman, in particular was known for poking fun at his wife: