The idea of magnitude coined by Leinster unifies several ideas of "counting" appearing in mathematics. The magnitude homology of metric spaces defined by Leinster-Shulman is a categorification of magnitude in a sense. To investigate precise meaning of magnitude homology, we study it for geodesic metric spaces by means of metric geometry. We show　several vanishing theorem for magnitude homology of geodesic metric spaces with an upper curvature bound. As a corollary, we obtain a slogan "the more geodesics are unique, the more magnitude homology vanishes".