What is "the sling"

The photo is a bit confusing, because it doesn't look at all as one would think it it should. Pull off from a roll of parchment a piece that is 8"wide. If you are not using a roll, cut a rectangle that is 8" wide and 12 - 13 inches long. Put a few dabs of butter on the bottom and two opposite sides of the square pan to help the paper stick. Lay the paper across so the two ends come up and over the sides on which you put butter. Press the paper into the pan and then butter the paper and unlined sides of the pan.

Hope this helps. ;o)

P.S. I'd like to understand why the paper in the photo looks the way it does.

It's described in Step 1 of the recipe. It's the parchment paper that you line the pan with and later use to lift the baked cake out of the pan.