The purpose of Haemon's speech in Sophocles' Antigone, as the questioner hints, is for Haemon to try to persuade his father Creon, the king of Thebes, to change his mind about the sentence of death that Creon has imposed upon Antigone. Antigone is Creon's neice and she also is pledged to marry Haemon.
Initially, Haemon's tactic with his father is to try to cast his argument in such a way that it sounds like the people of Thebes are making the argument and that Haemon is only reporting what he hears them saying. Haemon also draws on images from nature (trees that bend but do not break) and sailing (sailors who allow their sails to slacken in the face of a fierce storm). Thus, Haemon's main idea is that Creon should exercise moderation in his decree regarding Antigone.
The Chorus approve of Haemon's initial observations, but Haemon is put on the defensive when his father does not take his son's remarks as constructive criticism. Creon is immediately angered by his son's remarks and this puts Haemon on the defensive. Before Haemon leaves, at around line 751, the young man makes comments that foreshadow his own death later in the play.
This is a great question. The purpose of the speech to get Creon, his father to change his mind. Haemon knows that Antigone is innocent and does not deserve to die at all. So, what Haemon does is to relate what the townspeople are saying about the whole situation. At one point, Haemon says to his father that Antigone far from deserving death, actually deserves a golden crown for her loyalty to her brother.
The way he tries to convince his father to change his mind is through great tact. He acknowledges his love and respect for his father. He also acknowledges his youthfulness and lack of wisdom. At this point, he begins to rely what he has been hearing around the town, namely that Antigone is innocent.
When this does not work, he is more direct. This is where the tone changes. Haemon calls his father's actions unjust and bordering on insane. This, of course, angers Creon and hardens his resolve further.