What events causes Bilbo to recall the message in the runes?

Bilbo recalls the message in the runes after the company confronts the lonely mountain and realizes they need a way to open the secret door. The event that triggers his memory is the sight of the setting sun on Durin's Day, which aligns with the clues in the runes. This prompts him to reflect on the map and remember the significance of the moon letters, leading him to recite the message that reveals how to access the hidden entrance.