싸다 is wrapping or packing — food into a lunch box, luggage into a bag, or a filling wrapped in a leaf to eat, which is the very common 싸 먹다. -게 gives the manner, and here it is usually about quantity: 넉넉하게 싸서 (packed generously), 푸짐하게 쌀까요? (shall we pack a hearty amount?). Beware that an unrelated 싸다 means to urinate, which is the sense in the Brad Pitt line.