Manifold as a quotient?

Question

I'm trying to understand the following result from the book "Analytic Functionals on the Sphere" by Morimoto.

(Lemma 4.1 pg. 71) Suppose $n\geq 2$, $|\lambda| <r$. The group $O(n+1)$ acts transitively on $\tilde{\mathbb{S}}_{\lambda ,r}$. We have $$\tilde{\mathbb{S}}_{\lambda ,r} \cong O(n+1)/O(n-1)$$

Notation:

1. $O(n+1)=O(n+1,\mathbb{C})\cap GL(n+1,\mathbb{R})$ is the orthogonal group.
2. $\tilde{\mathbb{S}}_{\lambda} = \{ z\in \mathbb{C}^{n+1}\ : \langle z,z \rangle =\lambda^2 \}$ is the complex sphere of complex radius $\lambda$.
3. $\tilde{\mathbb{S}}_{\lambda ,r}=\{z\in\tilde{\mathbb{S}}_{\lambda}: L(z)\}$, were $L$ is the Lie norm.

My problem isn't so much with the proof, its more that I don't understand everything that's involved here. Could someone shed some light on how I should interpret this "quotient"?

On this question: Quotient group as a manifold one of the answers give a possible interpretation, in terms of lie groups. Is this the way I'm supposed to look at it? If so, could someone provide me a reference to this result?