Let $f(n)$ be the n-volume of the n-sphere. Then the natural thing to ask about is not $f(n)$, but $f(n)/2^n$; this is the ratio of the volume of the n-sphere to the volume of the n-cube in which it is inscribed. It's natural that this should be very small by a concentration of measure argument. The "typical" distance of a point in the n-cube $[-1,1]^n$ from the origin is a constant times $n$. More rigorously, pick a point uniformly at random from the $n$-cube; the square of its distance from the origin is $X_1^2 + X_2^2 + \ldots + X_n^2$, where $X_i$ is the $i$th coordinate, a uniform[-1,1] random variable. Thus $X_i^2$ has mean 1/3 and variance 4/45 (*), so the squared distance of a random point from the origin is roughly normally distributed with mean $n/3$ and variance $4n/45$. But points in the sphere are just those which have distance at most 1 from the origin, and these are quite rare.

(*) I'm not actually sure of this "4/45". In any case, it's a positive constant.