Patroclus is Achilles's closest friend during the Trojan war. Some readers speculate that the pair were either related in some way or that they had a romantic relationship as well as a fraternal one; either way, it is undeniable that they shared a deep bond.

Achilles withdraws from battle when Agamemnon takes his concubine away from him for no reason. He does not fight again until after Patroclus is killed. Patroclus takes Achilles' armor because he believes that the Trojans will be too afraid to fight if they see that the great Greek warrior Achilles has returned to the war.

Hector slays Patroclus believing that he was fighting Achilles. After this, Hector's fate is pretty much sealed, because Achilles's only goal will be to kill him. Achilles even has prophetic knowledge that his death will follow Hector's, but Achilles decides to make it his mission to kill him anyway. He returns to the battle in book 20 to avenge his friend's death.

I suppose you would say that the role of Patroclus is to be a sacrificial lamb in order to get Achilles to come back to the battle.

Achilles has been sulking in his tend because he was forced to give Briseis (the woman he had been given as a prize of war) up.  But then Patroclus gets killed by Hector.  Patroclus is Achilles's closest friend and Achilles is absolutely engraged by his death.  So that is when and why.  Achilles returns to battle when Patroclus is killed because he is so angry at his best friend's death.

If, by "when" you mean where in the book, it's in Books 18 through 20.