Pythia´s request to return home prior to being put to death is an important moment in the plot of the story. Lets remember why we came to this point:
Dionysius was the king of siracuse and was consider by the most part of its people a tyrant. According to some historian Pythias was conspiring against the regime but the official story said that he only was criticizing the way the king rule his kingdom and for that he was put into prison and punished with dead sentence.
The moment when the king was dictating the sentence is the moment where we have started above the answer. Damon, Pythias´s best friend, stood in front of the crowd and said that he would take his place in the jail to let his friend went back to his home town to close some unfinished bussines he had in there. During Damon´s time in prison he was the most happy convicted causing the surprise in the population that consider him as a fool firstly, but while the story advanced start to change their mind and considered him as a loyal friend. Finally the day on which the sentence must be done arrived and Pythias hadn´t show up yet, but Damon was even happier because his friend was not going to be killed. In the last moment Pythias arrived claiming that he have had a lot of problems on his journey but that he arrived on time to take his rightfull place. In front of such loyalty and friendship, Dionysius was astonsihed and took the decision of forgive the punishment of Pythias´s with the condition that they share the friendship with him.
So the right answer should be the last option: Damon demonstrates his loyalty by taking his friend´s place, which later causes Dionysius to reverse his decision to punish Pythias. I hope that this will be helpful for you.
