What is the content of [Zagiers proof](http://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2323918/fulltext.pdf)? 

What is the actual proof and why does it work? I am not sure I understand why,

1. there is only one fixed point, and
2. why that implies that the involution $(x,y,z) \to (x,z,y)$ proves the theorem.