There is a typo in the question: $n$ has to be large compared to $k$.

There are two proofs I particularly like:

1. A Morse-theoretic proof, probably due to Bott, can be found in Milnors book. Idea: consider the space of all paths on $S^n$ from the north to the south pole, which is homotopy equivalent to $\Omega s^n$. There is the energy function on this space. One does Morse theory: critical points correspond to geodesics (they are not non-degenerate, but Bott proved that a good deal of Morse theory works nevertheless). The set of absolute minima is the space of minimal geodesics connecting the poles. This is homeomorphic to $S^{n-1}$. All other critical points have index at least (rouhgly) $2n$. Therefore, the inclusion of the set of absolute minima into the whole space is $2n$-connected.

2. A spectral sequence proof (see Kirk, Davis, Lectures on Algebraic topology). Consider the homology Leray-Serre spectral sequence of the path-loop fibration $\Omega S^n \to PS^n \to S^n$. The total space $P S^n$ is contractible. Look at the shape of the spectral sequence; you'll see that for $k \leq 2n$ (again, only a rough estimate), all differentials out of the $E_{k,0}$-slot are zero, except of the last one, which gives an isomorphism $H_k (S^n)=E_{k,0}^{2} \to E_{0,k-1}^{2} = H_{k-1} (\Omega S^n)$. It is credible (but nontrivial to prove, this uses the transgression theorem) that this isomorphism is the same as the composition $H_k (S^n) \cong H_{k-1} (S^{n-1}) \to H_{k-1} (\Omega S^n)$ of the suspension isomorphism and the natural map $S^{n-1} \to \Omega S^n$. Thus the natural map is a homology isomorphism in a range of degrees, and by the Hurewicz theorem, this holds for homotopy groups as well.

There are two proofs I particularly like:

1. A Morse-theoretic proof, probably due to Bott, can be found in Milnors book. Idea: consider the space of all paths on $S^n$ from the north to the south pole, which is homotopy equivalent to $\Omega s^n$. There is the energy function on this space. One does Morse theory: critical points correspond to geodesics (they are not non-degenerate, but Bott proved that a good deal of Morse theory works nevertheless). The set of absolute minima is the space of minimal geodesics connecting the poles. This is homeomorphic to $S^{n-1}$. All other critical points have index at least (rouhgly) $2n$. Therefore, the inclusion of the set of absolute minima into the whole space is $2n$-connected.

2. A spectral sequence proof (see Kirk, Davis, Lectures on Algebraic topology). Consider the homology Leray-Serre spectral sequence of the path-loop fibration $\Omega S^n \to PS^n \to S^n$. The total space $P S^n$ is contractible. Look at the shape of the spectral sequence; you'll see that for $k \leq 2n$ (again, only a rough estimate), all differentials out of the $E_{k,0}$-slot are zero, except of the last one, which gives an isomorphism $H_k (S^n)=E_{k,0}^{2} \to E_{0,k-1}^{2} = H_{k-1} (\Omega S^n)$. It is credible (but nontrivial to prove, this uses the transgression theorem) that this isomorphism is the same as the composition $H_k (S^n) \cong H_{k-1} (S^{n-1}) \to H_{k-1} (\Omega S^n)$ of the suspension isomorphism and the natural map $S^{n-1} \to \Omega S^n$. Thus the natural map is a homology isomorphism in a range of degrees, and by the Hurewicz theorem, this holds for homotopy groups as well.