Haemon has learned that Creon has sentenced Antigone to death for her defiance, and he goes to Creon to try to persuade him to lift the sentence. Before he can get a word out, however, Creon guesses his purpose, saying:
My son, can it be that after hearing the final judgment concerning your betrothed, you have come in rage against your father? Or do I have your loyalty, act how I may?
Creon knows that his decision to execute Antigone is going to dismay his son, so he attempts to head off any arguments Haemon may make by demanding filial loyalty from him. He goes on to say:
This is the spirit you should maintain in your heart—to stand behind your father's will in all things . . . whomever the city may appoint, that man must be obeyed in matters small and great and in matters just and unjust.. . .There is no evil worse than disobedience.
By these words, Creon indicates that he is aware that his decision is, at the least, unfair, if not an outright injustice; but he argues that that shouldn't matter. The only thing that matters is that he is the ruler of the city, and of his family, and that Haemon, as both his son and his subject, should obey him in all things.
Haemon's response is to try to coax Creon to a different point of view: as ruler of Thebes, Creon surely has a responsibility to his subjects, and his subjects disagree with him on whether Antigone deserves to be punished. It is, in fact, Haemon's duty as Creon's son to let him know about the mood of the city, since ordinary citizens wouldn't dare tell Creon themselves.
I can hear these murmurs in the dark, how the city moans for this girl, saying: “No woman ever merited death less—none ever died so shamefully for deeds so glorious as hers. . . ."
The prevailing mood is one of dismay, which may fester into something darker if Creon does not take Haemon's words to heart.
Do not, then, bear one mood only in yourself: do not think that your word and no other, must be right. . . . You see how the trees that stand beside the torrential streams created by a winter storm yield to it and save their branches, while the stiff and rigid perish root and all?
Creon isn't persuaded; in fact, he's furious that his own son would disagree with him. He accuses Haemon of disloyalty and is insulted that someone so young would think their advice was worth anything. Haemon, who began the argument by stating that he loves and honors his father, is getting very frustrated with Creon's attitude. When Creon says Antigone is "wicked," Haemon retorts:
All the people of this city of Thebes deny it.
Creon demands to know whether "Thebes [shall] prescribe to me how I must rule," and Haemon has had enough:
See, there, how you have...
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spoken so much like a child!
What began as an attempt to reason gently with his father descends into a heated argument. Creon calls Haemon a traitor for "attacking" him, and says he is a "polluted creature, submitting to a woman." Haemon replies acidly:
You will never catch me submitting to shamelessness.
The two men are now shouting at each other, and any hope of agreement between them is shattered. Haemon storms out, warning Creon that by sentencing Antigone to death, he is also "destroying another"—foreshadowing Haemon's suicide later in the play.
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.