The word “emanated” means to come from, and in this context is used to reinforce the fact that the arena is the king’s idea.
The “semibarbaric” king feels that he has the perfect system of justice for his kingdom. He built an arena and installed two identical doors. Fate would ensure that the guilty person would choose the tiger, and the innocent person would choose the lady. To the king’s twisted mind, this was perfect fairness.
…[The king’s arena was] a structure which well deserved its name, for, although its form and plan were borrowed from afar, its purpose emanated solely from the brain of this man, who, every barleycorn a king, knew no tradition to which he owed more allegiance than pleased his fancy…”
The use of the word “emanated” reinforces that the arena comes from the king’s brain. The purpose of the arena, though built like other arenas "from afar," from elsewhere, is original to this king: this king alone has conceived of the idea to let Fate decided who is guilty and innocent by a forcing them to make a blind choice between two nondescript doors, one of which leads to death and presumed guilt, the other of which leads to life, reward and presumed innocence.
Further, the origination, the emanation, of the concept reinforces the idea that, in his kingdom, what he says goes. He does not care what anyone else thinks. He does not care about tradition. If he wants to do it, then it will be done. He is selfish, self-centered, and highly opinionated. He is not a good king.
It is interesting the way Stockton chooses words. His vocabulary is very sophisticated and specific. Each word is used for a purpose to create the desired effect. In this case, it reinforces the idea that the king is the sole originator of the idea and in full control.