It would be streets away by now, snatched, flying on the man’s arm as he ran; he would open it and inside he would find her keys, her identity card, her memories. Her entire life.”
-- Laure (The taxi had dropped her on the corner of the boulevard paragraph 4)

Importance: Laure losing her handbag is what sets up the plot of the book, with Laurent discovering the missing handbag and ultimately falling in love with Laure. This also explains why Laure was obsessed with the handbag, because it contained so many important items in her life.

What was the moral course of action now: to take it with him or to leave it where it was? Somewhere in the city, a woman had almost certainly been robbed of her bag and in all probability had given up hope of ever seeing it again.”
-- Laurent (Standing in front of the bathroom mirror paragraph 9)

Importance: Laurent discovering the bag is what sets him onto the path of discovering Laure. The fact that he cares enough to...