How can Homer's Odyssey be considered an allegory?

An allegory is defined as:

...an extended metaphor in which a person, abstract idea, or event stands for itself and for something else.

It says one thing on a literal level, but also makes a point or demonstrates a figurative meaning.

The term [allegory] loosely describes any writing in verse or prose that has a double meaning.

Homer's The Odyssey can be seen to be allegorical in many ways for the many tales we read about. As the story begins, Odysseus has been gone from home for twenty years. He went to fight in the Trojan War and served valiantly. On his return, his reputation is secure. Strong and smart, now he only wants to return home to his wife and son. For all the good he has done, and as deserving as he is, his journey home is not an easy one. The message Homer may be sharing is that Odysseus, for all his physical prowess and dedication in battle, is not as humble as he should be, and it is this lack of humility that costs him another ten years away from home. The stories Odysseus shares are exciting and show how smart Odysseus is and how dedicated to his men he is. However, he insults Poseidon, a powerful god, being arrogant towards Poseidon's son. For all the gifts a "great" man may have, what makes one truly noble is his ability to remain humble in the face of so much good fortune.

It begins in Book Nine when Odysseus' ship lands on the island of the Cyclopes (giants with one eye in the middle of the forehead). The giant, Polyphemus, captures Odysseus and his men. Odysseus charges the giant to observe the laws of hospitality:

'We...humbly pray you to show us some hospitality, and otherwise make us such presents as visitors may reasonably expect...Zeus takes all respectable travelers under his protection, for he is the avenger of all suppliants and foreigners in distress.’

Instead of showing his guests hospitality, the Cyclops proceeds to eat several of Odysseus' men. Odysseus is furious, however, he is also very clever, and he comes up with a plan to get away. Odysseus tells Polyphemus that his name is "Noman." Then he and his men get the giant drunk and blind him. When Polyphemus calls to his friends saying "Noman" is tormenting him, they tell him that of "no man" is tormenting him, he must be in trouble to cry about it—instead, he should pray to his father, the mighty Poseidon.

Odysseus (in the face of Polyphemus' suffering and criticism by his friends) shows his arrogance...

...I laughed inwardly at the success of my clever stratagem...

To escape from the giant's cave, Odysseus instructs his remaining men to strap themselves to the bellies of sheep and goats and then sneak out undetected when Polyphemus removes the rock that blocks their exit. They take all of the giant's sheep to the ship, as Polyphemus pursues them. Out on the water, Odysseus makes his biggest mistake by taunting the giant and showing his lack of humility, even though his men beg him to stop:

[I] shouted out to him in my rage, ‘Cyclops, if anyone asks you who it was that put your eye out and spoiled your beauty, say it was the valiant warrior Odysseus, son of Laertes, who lives in Ithaca.’

Odysseus continues to jeer at Polyphemus who then calls to his father, asking that Odysseus be punished. Poseidon honors his son's request, and it will take ten more years before Odysseus returns home, having lost all his ships and his entire crew.

This story proves the old maxim: Pride goeth before the fall.

What are some allegorical implications of Homer's The Odyssey?

Successive generations of thinkers read all kinds of allegories into Homer's Odyssey. One of them was the Neoplatonist philosopher, Porphyry, who interpreted the epic poem as a work of proto-Platonism. Odysseus regularly encounters a number of challenges on his journey, many of which are related to the sea. Porphyry interprets Homer's presentation of the sea as symbolizing matter, which for Plato and successive generations of his followers was inherently unreal. For them, the real was ideal, not material, consisting of unchanging, eternal ideas called Forms.

Yet only a philosophical elite were capable of apprehending such Forms. The vast bulk of humanity would remain trapped by a life of the senses, only able to understand what they could see, hear, smell, taste, or touch. The great hero Odysseus, however, is different. His epic wanderings are allegorized by Porphyry as an attempt to break free from a life of the senses and to ascend to the kind of wisdom which only those trained in the thought of Plato can attain.

Homer's epic poem The Odyssey was often read in allegorical terms during later periods in history, especially in the Christian Renaissance and middle ages.  Odysseus was often perceived as the virtuous man who is willing to struggle for virtuous goals in spite of all hardships. Likewise, Penelope was often perceived as the virtuous woman and virtuous wife who was loyal to her husband despite all temptations. In this respect, she was often contrasted with the adulterous Helen, whose disloyalty to her husband unleashed the Trojan War that is the main subject of The Iliad. The destruction of the suitors at the end of The Odyssey was often read allegorically as the inevitable destruction of vice by virtue.  Sir Philip Sidney, in his sonnet sequence Astrophil and Stella, alludes frequently to The Odyssey and clearly read the poem allegorically in Christian terms.

Renaissance Christians admired the kind of marital love expressed, for instance, in the following passage from The Odyssey, which describes the reunion of Penelope and Odysseus:

Her words stirred his heart to a greater longing for tears: and he wept, clasping his beloved, loyal wife in his arms. As welcome as the sight of land to the few surviving sailors, who swim to shore escaping the grey breakers, when their solid vessel driven over the sea by wind and towering waves has been shattered by Poseidon, who, saved from drowning, are overjoyed when their brine-caked bodies touch the land: welcome as that was the sight of her husband, as Penelope gazed at him, never unwinding her white arms from round his neck.  (Anthony Kline translation)