One of the biggest conflicts in Amélie is an internal one—do you stay in, or do you go out and see the world? Amélie's dad has the most trouble with this: he hasn't done any traveling in many, many years, because he's been too hung up on the death of his wife to do much of anything at all.
So Amélie gnome-naps his gnome, gives it to her flight attendant friend, and has her send her dad photos of the gnome abroad. The gnome visits Moscow, New York City, and Cambodian temples. Amélie's dad can't understand who would do this, and he's baffled every time he gets a new photograph. "He wants to see the world," Amélie explains.
And so, when her friend returns, Amélie puts the gnome back without a word. Her dad just gets up one day, and the gnome's there, on his dead wife's shrine, where he first put it. The gnome's mysterious journey prompts Amélie's dad to step out of his eternal mourning and into the real world, going on a trip for the first time.
Folks, it's pretty much a metaphor for what each and every character has trouble with: they all need to find a way to get out of their shells and start living.