In Plato's Apology, Socrates gives several reasons why weeping and wailing before the judges in order to attempt to avoid the charges or reduce the sentence would prove the charges against him. A crucial passage for understanding this is:
For if, O men of Athens, by force of persuasion and entreaty, I could overpower your oaths, then I should be teaching you to believe that there are no gods, and convict myself, in my own defence, of not believing in them.
In classical Greek rhetoric, persuasion was thought to occur by appeals from ethos (the character of the speaker), logos (reason), and pathos (by means of swaying the emotions of the audience). Socrates is making the argument that pathetic appeals are morally wrong.
He believes that it is the duty of judges to make impartial judgement based on reasoning. Pathetic appeals distort this process by asking judges to put aside justice itself and instead react based on emotions. This is precisely the opposite of seeking truth and wisdom.
Socrates believes that he has been divinely authorized to seek a certain type of wisdom He possesses a divine sign or daimonion that forbids him from acting in an impious way. Part of his religious duty is to seek knowledge and act as a gadfly to Athens, forcing others to seek truth as well by his questioning, For him to abandon this mission out of self-interest would be to disobey divine commands thus proving the charge of impiety.
Also, Socrates believes that the genuinely philosophical or pious do not fear death and so for him to plead with the judges to avoid death would also confirm the charge of impiety. Finally, manipulating the judges by emotional appeals would set a bad example for the youth watching the trial, confirming another one of the charges against him.