Why did the people in the community support the king's method of administering justice?
The common people had nothing to say about anything the king did. He was a feudal autocrat who ruled by force. The story opens with these words:
In the very olden time there lived a semi-barbaric king....
In that very olden time and place anybody who spoke a word against the king would be killed with the utmost cruelty. Such a person would probably not be put in the arena and forced to choose between the lady and the tiger. That was a more formal occasion. It was actually not until the publication of the American Bill of Rights that a country had guaranteed its ordinary citizens certain rights which could not be abridged by the government.
The Bill of Rights enumerates freedoms not explicitly indicated in the main body of the Constitution, such as freedom of religion, freedom of speech, a free press, and free assembly; the right to keep and bear arms; freedom from unreasonable search and seizure, security in personal effects, and freedom from warrants ...
If the people of the olden time in which "The Lady or the Tiger" is set had had such a bill of rights, they could have voiced disapproval of the king's methods under the articles guaranteeing freedom of speech, a free press, and free assembly. But the downtrodden and illiterate masses of that time could not even conceive of having such rights. They did not actually support the king's method of administering justice, or any of the king's other methods; they merely accepted them. The display of a man being killed and eaten by a tiger was one of the king's ways of reminding his subjects of what could happen if they got any wrong ideas.