In order to get the "Wooden Trousers" trophy in Burly Men at Sea, you need to choose to wear the barrels when given the prompt. Note that this will lock you into one set of possible endings and there are trophies tied to multiple endings. Use of cross-save and cross-load before this and any other trophy-related decision is recommended to shave down your overall completion time.
Here's the steps from when you enter the Whale, which is an unavoidable incident at the very beginning of the game:
-- Talk to the girls by the fire
-- Go all the way left until you see three barrels, interact with them
-- Interact with them again after the dialogue
-- Head right and go through the door behind your ship
-- Scroll right, get through the dialogue
-- Interact with the uvula, get through the dialogue
At this point the trophy should unlock.