He runs. He leaves his handkerchief behind (a detail that defines his character more than any sword ever could). But he does not run home. He catches up to the dwarves—breathless, panicked, and utterly out of place.
That is precisely why Gandalf chose him. the hobbit 1
The first journey of The Hobbit is a slow burn. It teaches us that a homebody can become an adventurer not by losing his love for home, but by discovering that courage is simply the willingness to be frightened outside your own front door. Bilbo will later find a magic ring, face Gollum, and earn his name as "Burrahobbit." But in the beginning, he was just a man who missed his kettle—and ran toward the mountains anyway. He runs