There are three different types of Chests on the Adventure board. Yuo claim their rewards the same way you do with any other reward from Adventuring - by ending your movement on a space with a Chest.
Chests always provide a choice of 1 out of three options:
Iron Chest
Choose one of the following 3 options:
• Tuck 1 card from your hand.
• Add 1 Supply to one of your Villager cards.
• Consider 3: Draw 1, Trash 2.
Gold Chest
Choose one of the following 3 options:
• Tuck 2 cards from your hand.
• Activate one of your Villager cards.
• Consider 5: Draw 2, Trash 3.
Ruby Chest
Choose one of the following 3 options:
• Tuck 3 cards from your hand.
• Search the Trash pile for any 2 cards. Tuck 1 of them and play
the other ignoring the cost.
• Consider 5: Tuck 1, Draw 2, Trash 2.
When your Adventurer ends its movement on a Chest space at the end of an Adventure path (with the reminder icon), return the Adventurer to the Home space at the end of your turn.