Poke a hole into the ball with the pin at as shallow an angle as possible, and add random markings with the pen (so the rotation is visible). Pick up the ping-pong ball with tongs, and immerse in the liquid nitrogen for about 10 seconds. Remove and release. Throwing it on the floor works well - it will hiss and spin and move horizontally in a random direction. In the video, I confined it using an extension cord so it stayed in close camera range, but it's better if you don't - much more fun to have it go whizzing off somewhere.
When the ball is immersed in liquid nitrogen, the air inside is chilled and the pressure drops (according to PV = nRT, and further yet because the oxygen will condense). The low internal pressure sucks liquid nitrogen in through the hole. On removing the ball, the nitrogen inside begins to boil as it heats up, and because gaseous N2 takes up about 700 times the space of liquid N2, the gas has to go somewhere, and it rushes out of the hole and turns the ball into a jet. Because the hole is at an angle, the resulting force causes the ball to rotate (spin) rather than translate (move). The steam is water condensing out of the air due to the cold gas, and so this experiment is more impressive in humid conditions.
Update 2016: The Backyard Scientist saw a gif of the video on Reddit and reproduced the experiment (he used liquid propane rather than liquid nitrogen) and estimated the rotation rate at 30,000 rpm.