Faxe is the Foreteller or Weaver of the Ariskostor Fastness and a leader in the Handdara religion.
Ai meets Faxe at the Fastness when he comes to have a prophecy foretold. As the Weaver, Faxe has the ability to tell the future by doing a seriously elaborate ceremony. According to Faxe, the art of Foretelling was created by the Handdara to "exhibit the perfect uselessness of knowing the answer to the wrong question" (20.111).
As a character, Faxe serves two purposes in the novel. First, his prophecy claims Gethen will join the Ekumen, and this convinces Ai to travel to Orgoreyn. So like the prophets of most other stories, he's a plot-mover.
Second, and more importantly, Faxe introduces the religion of Handdara to the reader. The Handdara religion is based on principles of inaction and accepting opposites—like the quote above. Ai eventually learns these two principles from Estraven, and both help him survive in the Gethnian wilderness.
We see Faxe again briefly at the novel's end. He's taken a job as councilman and is the first Gethenian to officially welcome the Ekumen to the planet.
And that's about it. We don't get much of a sense of that Faxe, or any of the Foretellers want. He doesn't appear to have much in the way of motivation; he just exists to help the plot along.