How do the dwarves open ths door to the dragons den?

In "The Hobbit," the dwarves open the door to the dragon's den by using a secret key and a specific method of revealing the door itself. The key, given to Thorin Oakenshield by his father, unlocks the door when the last light of Durin's Day illuminates the keyhole. As the sun sets, the dwarves find the hidden entrance to the Lonely Mountain, allowing them to confront Smaug, the dragon. This moment highlights the importance of timing and knowledge in their quest.