Joe's decision to commit suicide comes from the revelations in Anne's letter from Larry. When Joe finds out that his son, Larry actually killed himself because he could not bear to live with the crime that his father committed, Joe feels an overwhelming sense of guilt. He is consumed with a deep pain, a shame that he cannot face.
Everything that Joe Keller has worked for, stood for, wanted to be in that moment becomes a lie. He feels like a fraud, naked in front of his family, his wife, his son, Chris who can't bear to look at his father. Kate, secretly knew that Joe was guilty, but she did not know about Larry's suicide until Anne showed them the letter.
Joe realizes that he killed Larry, he precious son. He also killed the sons of other people, those men who were in the planes that crashed because of the faulty parts that Joe provided.
At the end of the play, Kate, sickened by Joe's suicide and by how she has been unable to feel alive since Larry went missing, tells Chris to live, to embrace life, to not run away from life. Kate comes full circle through tragedy, pain and grief to appreciate life.
Joe commits suicide in All My Sons because he found out the larry was dead. If he dident know that he was dead he would never have killed himself.