Why does the observable universe have this shape of two cones, when it's actually the same everywhere?

The answers so far talk about light cones, but I suspect you mean this type of image (from SDSS).

Each of the points in this picture is a galaxy and we are located at the center. The reason for the two blank cones is just that it's hard to map those areas - our Milky Way galaxy gets in the way!

So it's not that there's nothing there, it's just that it's a lot easier to see distant galaxies that are away from the plane of our galaxy, so that's where we tend to focus.