This paper is twisted, flattened, and crinkled rather than folded, and the resulting fungi are whimsical and delicate. He worked this up fast and it is actually quite simple to make.
There is so much amazing origami out there (like this and this and this). I am completely bewildered, knowing that each of those sculptures is made from a single sheet of paper.
If you're curious about origami, the crane is a great place to start. My girls (ages 6 and 7) can both manage a crane completely on their own, and they love it. (I would say 5 is a good age to start, as younger kids could be easily frustrated by the process.)
You can watch a clip from Between the Folds here. Check your local PBS listings to see the complete film, including how those mushrooms are made.