The curvature of spacetime around orbiting bodies is caused by the mass of those bodies, with the Sun's mass creating a distortion that influences the movement of the Earth. This is illustrated through an analogy of a bowling ball on a soft mattress, where the ball represents the Sun and the resulting depression affects the motion of a marble representing the Earth. The gravitational force experienced by orbiting objects is a result of their movement within this warped spacetime. Ultimately, the relationship between mass and spacetime curvature is fundamental to understanding gravity. The discussion emphasizes that mass is the key factor in causing the curvature of spacetime.