Socrates affirms that he has been consistent and just in his dealings with people throughout his life. He has never charged a fee for his teaching, has never kept secret any of thoughts, and has never refused to converse with anybody. The young men of Athens flock to him not because he preaches impious doctrines, as his accusers charge him, but because they enjoy hearing his cross-examinations: there is a good deal of amusement to be gained from watching the embarrassment of a pompous busybody. If these young have been corrupted by his bad influence, Socrates asks, why have they not since learnt the error of their ways and stepped forward to denounce him? On the contrary, Socrates cites the names of several of his pupils present in the jury--Plato among them--who are here in his support. Not only they, but their older relatives also take his side.
In summing up, Socrates alludes to the common practices of shedding tears, begging, and making mention of family members and loved ones in order to gain mercy. Though he has three sons of his own, Socrates scorns such methods for three reasons: first, it would be shameful and embarrassing, and such behavior would earn the scorn of foreigners; second, he would be asking the jury to consider facts that are irrelevant to the case at hand--if they are to deal justly with him, they should not consider such extraneous matters; and third, he would be asking the jurymen to break their oath of judging justly and impartially, a deed which would be highly impious. Of all places, Socrates would not like to appear impious when he is at court on charges of impiety.
In referring to the "corrupted" youth of Athens one more time, Socrates is not saying anything new, except that this time he identifies a number of his pupils by name. This is significant because Plato is one of the students mentioned. Most of Plato's dialogues, particularly the more mature works, are framed in a very complex manner. The dialogues are usually told by a third or fourth party who heard from a friend or acquaintance about a dialogue of Socrates' at which another friend was present. Not only does The Apology present Socrates' words verbatim, without any framing devices to distance the narration, but Plato makes a particular effort to point out that he was actually present at Socrates' defense. The purpose here might be to lend the retelling a certain authenticity: the author was present at the trial, and has copied down Socrates' speech word for word. It would be important for Plato to be able to claim such authority, as he wishes to acquit Socrates posthumously as much as possible.
Socrates' final remarks about not begging for mercy can be taken as either arrogant or ironic. On one hand, he is showing defiant bravery in a dangerous situation, while openly criticizing the normal practices of the law courts. On the other hand, this defiance could be read as very tongue-in- cheek. After all, Socrates alludes to his three children almost immediately after claiming that he would make no pleas based on his family life. This could be seen as a parody of a typical rhetorical maneuver. One's reading of this final passage depends on whether one prefers to see Socrates as the passionate and noble defender of a philosophical ideal, or as an ironic trickster who refuses to be taken too seriously.
This is arguably one of Socrates' most famous quotes, and in fact, living the examined life was the main cause of him sippin' on a killer cocktail. But, what does he mean? Why does he think that death is favorable option to living an unexamined life? What does the examined (or conversely, the unexamined) life look like? Another way to think about this is "Why does Socrates so willingly accept his fate?". Just food for thought.