What happens at the end is exactly what Jerry wanted to happen.  He wanted to die but not until he had an "intimate" connection with another human being.  Jerry wished to "matter."  He wished to know that his life, and in this case, his death, made an impact on another person.  Jerry could have committed suicide alone in his apartment and no one would have noticed.  He also could have killed himself in the presence of Pater for more of an impact.  BUT, ultimately, manipulating Peter to hold the knife (even if in a defensive mode) had the utmost effect on Peter's life.  Jerry tells Peter that he will never enjoy a Sunday on that bench again.  

After Jerry has completed his story about the dog, he sets about physically encroaching on his space. He tickles him, pushes him, punches him and tries to get him into a fight over the bench - Peter refuses.

Jerry then produces a knife - the first time in the play that we've been aware he possesses one! He throws the knife to the ground, and, again, challenges Peter to fight for his bench, hitting him and slapping him until Peter, finally losing his temper, picks up the knife. Then comes the surprise:

Peter holds the knife with a firm arm, but far in front of him, not to attack, but to defend.

JERRY
So be it.

With a rush he charges Peter and impales himself on the knife.

Jerry, mortally wounded, thanks Peter and tells him to run away, back to his house and his animals. Peter, after a few stumbling lines, does, screaming "Oh, my God!", words which, as the final lines of the play, are echoed by Jerry in "a combination of scornful mimicry and supplication". Jerry is left alone, dead, on the bench.