Why do Odysseus and his men burn an offering for the gods on the island of the Cyclops in The Odyssey?

While recounting the
tale of his travels to King Alcinous of Phaeacia, Odysseus says that when he and his men
discovered the cave of Polyphemus, the Cyclops, they first made a fire, next "made burnt
offering and [they then] took of the cheese and ate." Essentially, while they are waiting
for the cave's inhabitant to return, they made themselves at home in his home, eating and
enjoying the food they find. They are relying on an ancient Greek concept called
xenia, a concept which roughly translates to extreme hospitality. Any
travel during this era would require the traveler to rely on the kindness of strangers for food
and housing (and maybe more), and Zeus came to be the especial protector of travelers. Treating
travelers well became one way of heeding and paying homage to Zeus. Likewise, another religious
imperative dictated that, prior to feeding oneself, one would pour out and sacrifice some of
one's food to the gods. In order to claim their religious right to hospitality, sanctioned by
the gods themselves, Odysseus and his men really should adhere to other religious rules, such as
burning an offering of food to the gods, when appropriate.