In "The Lady, or the Tiger?", why are the public trials in the arena popular with the people in the kingdom?
In a society like that in this story that is dominated by a brutal king who uses the arena to exact justice on those that he perceives have broken his laws, the presence of the people of the kingdom is probably not voluntary but mandatory. Not that the people are forced into the arena to observe the killings that appear to be routine, but because the King uses the arena as a system of justice.
As a system of justice, the punishment that is rendered by the King serves as a strong message to any subject that might consider breaking the law in the kingdom. Whether it is a civil or moral law. The gruesome deaths that take place in the arena act as a deterrent to others who might consider a life of crime. They dare not choose this path, the King shows no mercy.
Another reason that the people all gather in the arena is because they have no other forms of entertainment at this disposal. Most of the people are commoners with little money and no means of providing for other forms of entertainment. The arena provides an opportunity for the common people to be entertained and warned at the same time.