Why is it important that the accused people come out underneath the place where the king and his party sit in the arena in the story "The Lady or the Tiger?"

The location of the king’s party is important because the princess is able to secretly signal her lover.
It is important that the king and the accused can see each other, because the king feels that fate is at play in his system of justice.  He is not the one condemning the accused or pardoning him.  That is done by fate.  The accused person chooses between two doors.  One kills him and the other sends him to his wedding to a beautiful maiden.

Directly opposite him, on the other side of the enclosed space, were two doors, exactly alike and side by side. It was the duty and the privilege of the person on trial to walk directly to these doors and open one of them. 

However, it probably never occurred to the king that someone in his party might find out what was behind each door and signal the accused.  When his daughter’s lover was sent into the arena, she took it upon herself to find out what was behind which door.  She used her position to signal him.

Her right arm lay on the cushioned parapet before her. She raised her hand, and made a slight, quick movement toward the right. No one but her lover saw her. Every eye but his was fixed on the man in the arena.

He knew that she would find out, and he knew she would signal him.  The interesting thing about this story’s ambiguous ending is that we do not know what she signaled him.  She might have sent him to his death, because she did not want another woman to have him.  However, she might have also decided she could not live without him.  We do not know what the signal was, we only know that there was a signal, and he saw her signal.