This quote is in reference to the King.  The autor is lettinng us know the character of the king.  He was a man who asked for counsel from no one. This is what is ment by "self communing."  He thought about what he wanted to be done and it was done.  No trial, no jury, simply the king decides that the accused will open one door or another.  If the man is guilty, he will be eaten by a tiger.  If he is innocent, he will marry there on the spot.  

  "He was a man of exuberant fancy, and, withal, of an authority so irresistible that, at his will, he turned his varied fancies into facts. He was greatly given to self-communing, and, when he and himself agreed upon anything, the thing was done. "

The story was originally published in 1882. The author Frank R. Stockton's style of writing may seem old-fashioned by contemporary standards. He is simply being facetious when he says of the king, "He was greatly given to self-communing, and, when he and himself agreed upon anything, the thing was done," it is a whimsical way of saying that the king was an autocrat and never had a need to consult anyone else about his edicts. He might be compared to King Henry VIII or even to the Roman emperor Caligula, or to another Roman emperor, Nero, who thought of the idea of feeding Christians to lions and tigers in arenas seating many thousands of spectators.

He was a man of exuberant fancy, and, withal, of an authority so irresistible that, at his will, he turned his varied fancies into facts.

Stockton's tongue-in-cheek humor is deliberately intended to make light of a gruesome subject. If the author were to describe the defendant's ordeal in straightforward prose, it would only make it seem vile and loathsome. But using a rather fanciful and somewhat antiquated prose style, as well as setting the story far back in time, have the effect of softening the grim actuality. The story would not have been as popular as it has been over all these years if the tone had not given it a sort of fairy-tale quality. In fact, the author seems to have chosen exactly the right stylistic approach as well as the right ending.

It is noteworthy that while the king is an ogre and is responsible for the lover's plight and his daughter's distress, the king himself is the least important character. He remains mostly in the background, while attention is focused on the princess and her lover. 

Simply put, it means the king answers to no one.  He only consults with himself and it only takes he and himself (a literary play on words that means  he and he alone) to make a decision.  This undoubtedly goes to the barbaric side of his nature.  When the author tells us that it is the daughter of the king who makes the decision in "The Lady or the Tiger?" as to who will come out of the door, we may gain insight as to what will be on the other side of the door.

But as Frank Stockton put it, we have to leave that with our noble reader.