Why are Odysseus and his men interested in the Cyclops' cave?

Odysseus and his men are interested in the Cyclops' cave because they are curious and are hoping to receive the gift of hospitality from the occupant of the cave. In ancient Greek culture, hospitality (or "xenia") was an extremely important practice, and it was expected that hosts would offer food, shelter, and gifts to their guests. Also, they may have been seeking provisions or assistance as is customary when visiting new lands.

Upon arriving at the island of the Cyclops, Odysseus wanted to meet the inhabitants and see what they had to offer. They noticed the cave was well-stocked with provisions like cheese and livestock, which indicates that the cave's occupant had an abundance of resources. As they were on a long and challenging journey, they were naturally drawn to the prospect of replenishing their supplies and perhaps finding an ally in the Cyclops.

Unfortunately, their encounter with Polyphemus, the Cyclops in question, does not go as hoped. Polyphemus does not respect the laws of hospitality and instead poses a grave threat to Odysseus and his crew, ultimately consuming several of the men before Odysseus devises a plan to escape.