I made a pair of these chairs a few years ago while I was in school, and had access to a nicely-equipped shop and, most importantly, a laser-cutter. The tennis balls you see are not glued in place; they are held only by different-sized holes in the top and bottom sheets of plywood. The holes on the bottom are smaller so the balls don't push through when you sit on it. To create the contoured effect, the sizes of the top and bottom holes vary in specific ratios. The balls can still freely rotate and some of them can come all the way out. That said, it is much easier to make without all the contouring, but decidedly less comfortable. I made prototype version in which the balls do not contour, and will include some pictures of that chair at the end of the instructable.
I have unfortunately lost some of the construction photos, so, in lieu of illustrations, have tried to make the written part as detailed as possible.
Everyone always asks me if it's comfortable. Yes. The key here is that the balls are not glued in place or screwed through somehow, so they can flex and deform. With the body distributed over fifty of them, no single one makes a pressure point; the slope on the seat pan, and the U-shaped contour on the back match butt and back as closely as possible, giving spine and tailbone some breathing room.
The fact that they are not fixed in place also means that they shrink and expand from their holes with changes in heat, humidity, and altitude, as the air trapped inside the balls changes in volume.