In your analogy, and I can't stress that it is an analogy enough, the surface of the balloon is representing 3d space. We don't see the interior of the balloon because it's not the universe it's the shape that the universe is curved around.

What we actually observe is that distances on a cosmic scale are all getting longer at a range proportional to their length. This is distinct in some ways from observing objects moving away from us, specifically it isn't restricted by the speed of light as the objects don't really have the velocity that the changing distance to them implies.