Rem was part of Project Seeds, humanity's last-ditch escape from Earth after the planet became completely uninhabitable. When the ship encountered a young Vash and Knives stranded on a hostile world, most of the crew wanted them dead—but Rem intervened, refusing to treat the brothers as monsters. She showed them genuine compassion instead.
She spent her downtime in the ship's holographic Eden, a simulated recreation of her old Earth home stripped of war, theft, and hatred. Vash became deeply attached to her and adopted her worldview wholesale. Knives loved her too, though he drew very different—arguably opposite—conclusions from what she taught.
Rem is basically the entire moral compass of Vash's adult life. When she told him "No one has the right to take the life of another," it stuck. That's why he refuses to kill, period. The iconic red coat he wears is also her legacy: it was inspired by her favorite flower, the red geranium, which symbolizes unwavering determination. The coat functions as a constant reminder of both Rem herself and the promise he made to her—to never take a life and to help others whenever he can.
Content compiled by AnimeList.moe from publicly available sources.