In Plato's Apology, Socrates chose death above living without his beliefs. It followed Greek thought that death was preferable to banishment from their homeland. Socrates does not mind being sentences to dead, because he is an old man in his seventies who believes he is going to a better place.
The jury offers to let Socrates off on the, "condition that [he] spend no more time [practicing] philosophy." (p. 32) The jury are mocking him because any true Greek would never give this a second thought. Socrates replied that he would "obey the god," (p. 32) who set him on his course in life; rather than comply with the jury. In the Greek epic, The Odyssey by Homer, Odysseus angers Poseidon, but he doesn"t kill him. Instead, he traps him on an island for years, which is a fate worse than death for Odysseus. And if Socrates gave up practicing philosophy, that would be a fate worse than death as well. .
Socrates is an old man, who has lived long than most people usually do. He knows that this trial is just a formality and his death has already been decided. They are letting him defend himself as part of this formality. But Socrates has no qualms with going to the after life. In fact, he is happy to go to a place were he could continue his philosophizing and spend is time with "Palamedes and Ajax to compare [his] experience with theirs and examine people there as [he does] here, as to who among them is wise." (p. 42) Socrates is not scared of crossing over from the mortal plane, but welcomes it.
Socrates would rather be put to death than be living a lie, and thus he chooses to shun the jury's deal. He is an old man, and won"t be alive for much longer in any case. Wherever he goes, Socrates believes he will be happier there.
"Now the hour to part has come. I go to die, you go to live. Which of us goes to the better lot is known to no one, except the god. " (p. 42).