Why were the shows in the arena important to the king's people?
In Stockton's "The Lady, or the Tiger?" it's not just the trials that the people like, but the whole structure of the arena as well. The text says that the king built the arena not just for violent entertainment, "but for purposes far better adapted to widen and develop the mental energies of the people." That is to say that the king wanted to also mentally challenge his subjects, not just to provide an event to waste time. The structure of the amphitheater also has "encircling galleries, its mysterious vaults, and its unseen passages." This seems like a veritable funhouse for those seeking a mental challenge as well as entertainment.
As far as the shows, or trials, are concerned, the people take great interest in them because of the "element of uncertainty" that goes with each one. As a result, they become very popular among the king's subjects. Attendees never know if they will see someone violently torn to bits by a tiger, or married to a mysterious young maiden. The audience would play along with the scene, too. If the accused died, the whole audience, along with hired mourners, would show remorse and bow their heads as they walked home. If the accused got married, a great party would erupt and the audience would cheer and support him in his new joy. Hence, the populace is entertained with either outcome; and, the intellectuals of the community don't argue because the king would simply say that the accused had fate completely in their own hands. In this way, all of his subjects are satisfied and the arena becomes a focal point in the kingdom.