Permit me to suggest another "intuitive" approach, which hardly uses any calculations, just some basic Arithmetic (that means: Combinatorics)
Well, it is a fact that even for big numbers of n (dimension) the unit n-sphere inscribed in the unit n-cube is still the LARGEST possible. Its diameter is always equal to the "side length" of the cube, and the sphere's surface touches every "face" of the cube (as "Face" for an n-cube, is to be considered of course an (n-1)-cube).
So, why the "shrinkage" of the sphere for higher and higher dimensions? It is simple a matter of Arithmetic. The sphere always occupies the "central area"/center of the n-cube, but there is not much "center" left, as n goes to infinity. Most of the cube's volume "escapes" (centrifugally, sort of) towards the corners/"vertices". For example a 100-cube has 200 "faces" but $2^{100}$ vertices. Let's consider that we produce the "diameters" of the cube, namely all the straights that pass through the Center of the cube.
There are two main types, as far as "length" is concerned. The smallest, let's call them "good" or "short" which start from the center of a face, pass through center C of the cube and end on the center of the opposite face. For the usual 3-dimensional cube that prescribes a unit sphere (r=1), this length of the short diameter is 2 (since 2 is both the diameter of the sphere and the edge length of the cube). That shows that a "good/short" diameter is spreading entirely (its whole length) INSIDE the sphere. A "bad/long" one from the other side, are these diameters of course which start from a vertex---trough center C---to the opposite vertex. For an n-cube with side length=2 , the long "bad" diameter has length = $2\sqrt n$. Thus, for example for a 100-cube the long/bad diameter has length equal to 2*10=20. We notice that just 1/10th of this length lies in this case inside the sphere!
Moreover, there are 100 "good/short"/full" diameters, but $2^{99}$ long ones! ("empty/bad"). The above argument "by induction", may be perhaps not so mathematically strict, but it is logically very powerful, I think.