For me, this is pretty much the core argument that makes the paradox vanish: For instance, naive students of probability may expect the average of a product to equal the product of the averages (OP pdf page 4) It's not too hard to convince yourself that you can't stir numbers together at random and get away with it, even if it nearly looks sensible at first. The language says it all: "average of products" and "product of averages".
The Vector Interpretation in the WP article gives a pretty decent visual explanation. If you find the maths a bit off putting just squint past the notation. You can pretty much replace all occurrences of funky symbols and vector with the word "line", accept that you can add these special lines together and you should be able to get the gist of what is going on.
I found this unbelievable so I wrote a program to generate random contingency tables and search for a double reversal. When I found one I wrote a little story around the numbers.
Wikipedia got the joke: "Suppose two people, Lisa and Bart, ..."