What is this design style called in which lots of simple polygons make up an image?
While we could trace the aesthetic back to various art and design styles such as the aforementioned cubism, there are some more recent terms such as low-poly or faceted illustration. Tim Reynolds was one of the earlier users of the style.
It can be done via 3D rendering (where it originates...essentially reducing the polygons used to render a complex object), or 2D filters (or just by hand). There's an App for it too: Dmesh
Related to it would be voxel style which, instead of connected planes to create a 3D shape, is used individual 3d shapes (such as in Minecraft).