After his visit to the underworld in Book XI, why does Odysseus return to Circe's island?
The shade (ghost or spirit) of Elpenor, one of Odysseus' men whom he spoke to in Hades, begged Odysseus to give him the proper funeral rites. The Greeks believed that a soul suffered and could not be at rest unless it was properly buried (this was the impetus for the plot of Antigone). Since Odysseus had promised to give him all the proper funeral rites, he needed to go back to Circe's island to do so.
Then, when the child of morning, rosy-fingered Dawn, appeared, I sent some men to Circe's house to fetch the body of Elpenor. We cut firewood from a wood where the headland jutted out into the sea, and after we had wept over him and lamented him we performed his funeral rites. When his body and armor had been burned to ashes, we raised a cairn, set a stone over it, and at the top of the cairn we fixed the oar that he had been used to row with.
Not only was Odysseus fulfilling a promise to his dead friend, but he was discharging a sacred duty. It was understood, among the ancient Greeks, that men who fought together would bury each other if they fell. For Odysseus not to bury his friend would be more than just a broken promise; it would be the dereliction of a sacred duty. For Odysseus to refuse to bury Elpenor would be to appear very untrustworthy.