This is actually slightly different issue.
Regardless of the dimension, the volume of n-ball is V_n(r) = c_n * r^n, where r is radius, and c_n is a constant depending on n. The grandparent observation is that surprisingly enough, c_n goes (rather quickly) to 0 as n goes to infinity, so for unit n-ball, that is, a ball of radius 1, the volume is exactly c_n, which is very small. This is surprising to us, because in familiar case of n = 3, c_3 = 4/3 pi, which is moderately large.
As for the mass being concentrated around the peel, suppose we have an orange of radius R+e, and its peel has thickness of e. Then, the ratio of volume of the peel to the volume of the whole orange is exactly:
(V_n(R+e) - V_n(R))/V_n(R+e) = 1 - V_n(R)/V_n(R+e) = 1 - c_n R^n / c_n (R+e)^n = 1 - R^n/(R+e)^n = 1 - (R/(R+e))^n
Now, since R/(R+e) Note that the peel argument works just as well with "square" oranges -- they also have most of their mass concentrated around the peel, but contrary to round oranges, their whole mass does not go to 0 as the dimension increases. To see that, note that the volume of n-dimensional square with side of R is exactly R^n, and if you do the above computation, it's exactly the same (note that the constant c_n cancelled out anyway).
In this sense, your comment and the grandparent ones are about two different phenomenons -- grandparent is talking specificly about the geometry of the sphere in L_2 norm, while you are talking generally about n-dimensional volumes.