In this particular analog to the expanding universe, all of 3-dimensional space is within the surface of the balloon. What is inside and outside the 'balloon' is the coordinate which is at right-angles to the space axis, which is of course, the dimension of time. The balloon analogy is meant to show, for a CLOSED universe, that its space-time is curved through the 4th dimension to form a closed geometric shape. There is no additional space outside the cosmological 'balloon' because all of our 3-d space is within the 'hypersurface' we see in the balloon's surface.