By Hurewicz (n-1)-connected implies vanishing of the first n-1 homology groups. Since the manifold is closed and (by simple connectedness) also orientable, we have H_n=Z. Of course the higher homology groups vanish. Thus the manifold is a simply connected homology sphere, hence by Hurewicz' converse a homotopy sphere.

By Hurewicz, (n-1)-connected implies vanishing of the first n-1 homology groups. Since the manifold is closed and (by simple connectedness) also orientable, we have $H_n={\mathbb Z}$. Of course the higher homology groups vanish. Thus the manifold is a simply connected homology sphere, hence by the converse of Hurewicz a homotopy sphere.