The circle and the north pole (or wherever the origin of the stereographic projection is) span a 3-dimensional subspace generically, such that the restriction to this subspace is the 2-dimensional stereographic projection. If the circle goes through the north pole, then it is actually sent to a line under stereographic projection, and this is in some sense a reduction to the 1-dimensional case.

Yana Mohanty has a nice proof that stereographic projection sends circles to circles.

A more sophisticated approach is to notice that stereographic projection is the restriction of inversion through a sphere orthogonal to $S^n$ in $R^{n+1}\subset S^{n+1}$. Then one needs to see that inversions send circles to circles, or more generally that Mobius transformations of $S^n$ do. The group of Mobius transformations of $S^n$ is $PO(n+1,1)$ or $Isom(\mathbb{H}^{n+1})$, the isometry group of hyperbolic $n+1$-space. This groups preserves the cone $x_0^2+x_1^2 +\cdots - x_{n+1}^2=0$. The sphere at infinity (in the projectivization) of this cone is $S^n$, and the action is by the Mobius group. A circle is the intersection of the projective closure of a 3-dimensional subspace with the sphere at infinity. Since $PO(n+1,1)$ consists of linear transformations, it permutes 3-dimensional subspaces of $R^{n+1,1}$, and therefore sends circles to circles in the projectivization.