What is the significance of the Trojan War in the Odyssey by Homer?
The Odyssey by Homer is the story of the return of Odysseus from the Trojan War. Thus the significance of the Trojan War is that it, especially as recounted in The Iliad by Homer, acts as a sort of prequel to The Odyssey. Many of the events that happen in the process of the return of Odysseus are dependent on prior events in the Trojan war and the alliances and hostilities among people and gods ensuing from that war. Both Athena's patronage of Odysseus and Poseidon's enmity to him spring in part from the events of the war.
The Odyssey is the aftermath of the Trojan War, as recounted in the Iliad. All of the events of the Odyssey are dependent on the Trojan War.
Twenty years before the start of the epic, Odysseus leaves for the Trojan War, which takes ten years to win. The Odyssey fills in the gaps of the ten years after that, and the reasons why it took him so long to return home to Ithaca again.
The situation at Ithaca is dire. Suitors for Odysseus' wife Penelope have overrun the palace, and thing do not look good. Their son Telemachus is trying to find more information about his father. All of this is because the Trojan War took Odysseus away in the first place.
Meanwhile, Odysseus is fighting to try to get home. Because of the events of the war, he has the goddess Athena on his side, but the god Poseidon is against him.
Throughout the entire epic, the events of the war continue to be mentioned again and again. Stories of the war get brought up with Nestor at Pylos, with Menelaus and Helen at Sparta, and with Alcinous and Arete at Phaeacia. They tell of the battles/struggles, the final stratagem, and the homecomings of the various Greek warriors.