Ah, ok, I will try to appeal to symmetry while giving geometric intuition. I think the best way to visualize this (and explains a little what exactly is going on) consider cutting a plane through the hypercube through diametric corners. For instance, (2,2,2,...,2) and (-2,-2,-2,...,-2). This will reduce down to a 2-dimensional problem but things are no longer square. The vertical edges of the quadrants are 2 but the horizontal lengths are 2sqrt(D-1). To see this, calculate it for 3-dimensions as you can still draw it out. By construction, the circle inside the quadrant kisses the top and bottom and is centered horizontally. That is, there is a gap between the edges of the circle horizontally and the quadrant. Again, to visualize, picture how the plane is cutting in 3-dimensions. Now, since you seem to accept that the circle is on the origin in two dimensions, the circle is on the origin in this plane. Repeat this procedure for all diametric corners and, by symmetry, the inner sphere is always centered at the origin. For completeness, because the inner sphere is centered at the origin, the radius must grow along the diagonal. So, we can use our squished quadrant again to calculate the radius, r, of the inner sphere. It is going to be sqrt(2^2 + (2sqrt(D-1)^2)-2 = 2r, where the subtraction is the outer sphere. Some arithmetic gets you to r = sqrt(D)-1 which is of course what they got in the post.

Hopefully this was the correct amount of words and equations that you can reconstruct this on paper. I think doing it this way shows you that the outer spheres get increasingly smaller relative to their containing "quadrant". For me, it also elucidates that the final drawing in the post is very misleading because the hyperspheres don't spike like that. Rather, they look more like 4-leaf clovers and that is what allows it to escape the cube in the additional "horizontal space" of the aforementioned quadrant.