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Okonkwo hangs himself because he is unable to adapt to the new society.
He knows he will be imprisoned, (and probably executed) for killing the messenger. He is also a proud man, so prefers to take action into his own hands. By killing himself, he controls his own destiny instead of letting the Europeans control him. He also knows that his clan will not fight the Europeans.
Some have argued that he hangs himself because he is ashamed of his village, but if that were so, than he probably would not have killed himself, which is a sacrilege to his village.