It is obvious to us that the ancient Greeks valued hospitality because of how crucial it is in The Odyssey. When Odysseus and his men land on the island where the Cyclopes live, he wants to remain in Polyphemus's cave to see if the Cyclops "might offer gifts": giving a guest-gift was a common practice of those who offered such hospitality to strangers. However, the monster does not make such an offer and, instead, actually eats several of Odysseus's crew. When Odysseus has his remaining men escape, he shouts back,
"It was also destined your bad deeds should find you out, audacious wretch, who did not hesitate to eat the guests within your house! For this did Zeus chastise you, Zeus and the other gods."
The Greeks believed that Zeus protected travelers, and so offering hospitality became a sort of religious imperative to them; to serve Zeus, you help travelers. Here, Odysseus claims that Zeus allowed Odysseus to blind the Cyclops and escape his island because the monster failed to offer hospitality.
We can also see the importance of hospitality in Alcinous's treatment of Odysseus. The king and queen welcome him into their home, feeding and clothing him, even giving him a place to sleep for the night, before they even ask his name. It is only after quite some time spent feasting, listening to music, and so forth, that Alcinous finally says,
"And do not you, with wily purpose, longer hide what I shall ask; plain speech is better. Tell me the name by which at home your father and mother call you . . ."
It's not as though they know it is Odysseus, the great war hero. They only know that he is a stranger in need, and so they meet those needs and then some. We see, then, how important hospitality is to this culture.