"Sokrates is guilty of criminal meddling, in that he inquires into things below the earth and in the sky, and makes the weaker argument defeat the stronger, and teaches others to follow his example."
This "informal" charge is inadvertanly brought on by Aristophanies and his play "The Clouds." The character "Sokrates" in that play is precisely this kind of nature philosopher! The problem for the real Sokrates is to convince the jury that he bears little resemblance to the character of the comedy.
Having claimed that he is not guilty of either set of charges, Sokrates offers an alternative account of why Meletus, Anytus and Lycon have brought these charges against him.
Sokrates discovers that none are wise concerning virtue, and in the process of his investigation he makes enemies by publically demonstrating that those who claim to be wise are not. Thus, the real reason he's been brought before the court is because he's made enemies in all parts of the community by exposing those who pretend to be wise when they are not.
"Sokrates is guilty of corrupting the minds of the young, and of believing in deities of his own invention instead of the gods recognized by the state."
Both of these arguments against Meletus are excellent examples of Sokrates' method of argument called 'elenchus' in Greek, or "cross examination." Notice that Sokrates himself does not put forward any specific claims, but merely uses the claims of Meletus to demonstrate an inconsistency in the prosecutor's case. This seems to be the way Sokrates always argues with his opponents. If we look at other dialogues from Plato's Socratic Period, we find Sokrates using the same style of argument. This allows him to claim that he does not have significant wisdom on one hand, while at the same time also claiming that his interlocutor doesn't have significant knowledge either!
The elenchus is quite different from what is commonly called the "Socratic Method" which we will see demonstrated in Plato's dialogue the Meno. There we will see Sokrates guiding his interlocutor to the correct answer through a series of leading questions. This is sometimes also referred to as the dialectic method of argument. The dialectic method, however, is probably a platonic invention and not part of Sokrates' style of Philosophy. It is therefore a misnomer to call it the Socratic Method.
In the courts of Athens, the prosecution would suggest one puninshment and the defense would offer a counter punishment. Then the jury would decide between them. Meletus, for his part, has put forward the death penalty, so it's Sokrates' turn to offer a counter penalty.
This is an important, and very interesting, part of the trial. Sokrates has been found guilty by a narrow margin, which likely means that he can sway the jury to leniency in the penalty phase. So Sokrates begins his address to the jury by suggesting lifetime maintaince at the Prytaneum. The Prytaneum is the place in Athens where former victors of the Olympic games received daily rations of food, this is the reward they received from the city for their victory. In essence, Sokrates has suggested as his penalty free meals for the rest of his life! It is clear that he doesn't think he will get this because he quickly goes on to explore banishment, which he rejects, then a monetary fine. We have to ask ourselves, why did Sokrates do this? The effect it has on the jury is obvious; when the second vote is taken, some of those who had voted to acquit Sokrates a moment before, now voted to put him to death.
One possible explaination, the one given by Xenophon in fact, is that Sokrates is trying to get himself killed. But this seems inconsistent with everything else we know about Sokrates. I think a better explanation is that Plato is showing us that Socrtates is a flawed character. He is not perfect, and his sarcasm sometimes gets the better of him. The matter is open for interpretation, however, and you must struggle with this question and come to an answer for yourself.
In other words, Philosophy is not a pastime, or sport, or mere entertainment. It is something one does with the whole being. It is, for Sokrates at least, the most important way to live one's life.