In the Apology, Socrates reasoned that he should receive the death penalty, not a light decision by any means. Socrates was imprisoned and essentially forced to recant his beliefs.

In light of this, Socrates believed that, in order to remain true to his beliefs, he should receive, and would accept, the death penalty. His conviction was so resolute in his beliefs that to deny them would be a worse fate than death. He writes his book outlining his core beliefs and decrying the injustice of the situation, while also accepting that the gods controlled his fate and that following his convictions would be worth more than his life. He accepted that this was not a fair trial but knew that justice would eventually be served, and he would ensure that his conscience was clear upon his death.

It's not so much that Socrates wanted to be executed; he didn't have a death wish, or anything. It's just that he'd lived his whole life in Athens, always abided by its laws and so naturally thought it would be unreasonable to start complaining about those laws simply because he was going to be put to death by them.

Socrates had many loyal friends, and some of them put together a plan that would've allowed Socrates to escape from prison and avoid execution. But Socrates wouldn't hear of it; he was determined to stay put and take his punishment. He'd spent his whole life teaching others that the most important thing in life was how to prepare for one's death, and that the manner of a man's death should reflect how he'd lived his life. And as Socrates had lived a life in strict fidelity to the laws of the state, he thought it only right and proper that he should die by them, too.

Actually, the question should not have been changed. Plato wrote a fictional speech which he put in the mouth of Socrates. This is not a verbatim transcription. The Platonic and Xenophonic accounts of the trial differ.

He could not have opted to be acquitted -- any more than a defendant in any other law court is given a choice of aquittal. Instead, as was standard Athenian legal procedure, after he was condemned, he was permitted to suggest an alternative penalty. This is not acquittal. Normally, he would have been expected to suggest the counter-penalty of exile, or a monetary penalty, which, for the reasons you state, he does not do. Instead, he suggests that he be supported for the rest of his life by the state, a suggestion that mocks the process of the trial. The counter-proposal is not accepted and he is therefore condemned to death.

First of all, please realize that it is Socrates, not Plato, who is arguing that Socrates should be executed.  I have changed your question to reflect that.

Basically, Socrates insisted on being executed because he thought that was the only choice he had if he did not want to betray his principles.  He could have, for example, opted to be acquitted if he would give up his teaching, but he felt that his mission in life was to do just exactly that -- to go around teaching.  Therefore, it would be better for him to die than to give up the mission that he had been given by the gods.

Socrates believed that what he was doing was important.  He believed that it was important for him to go around goading people into doing right.  He felt that this mission of his was so important that he was willing to die rather than to give it up.  This is why he argued that he should be executed.

Why was Socrates given the death penalty?

When Socrates was tried and convicted (and sentenced to death) there were two charges against him.

First, the Athenian authorities claimed that Socrates refused to worship the Greek gods.  This was completely untrue -- Socrates was always diligent in performing the required rituals, even though he did not necessarily believe in them.

Second, he was charged with corrupting the youth.  The basic idea here is that his questioning was turning them against the values and the political system of the city.  The general reason for this was that he kept criticizing democracy.  It was also in part because a couple of people who had been his students became dictators in Athens.

You can also say that he was executed because he was stubborn.  At his trial, he refused to apologize or show remorse or even accept exile.  In that way, he sort of forced them to kill him.