Let $\pi: S^{2n+1}\rightarrow \mathbb CP^n$ be the quotient map. Then $\pi^{-1}(p)$ is diffeomorphic to $S^1$ since $\pi(e^{i\theta}z)=\pi(z)$ for all $\theta$ where $z\in S^{2n+1}$.
Now let's consider the fibration $S^{2n+1}\times S^{2m+1}\rightarrow \mathbb CP^n\times \mathbb CP^m$ which has fiber the torus $S^1\times S^1$. How can we now conclude that $S^{2n+1}\times S^{2m+1}$ is a complex manifold?
I don't exactly know what the author meant by the paragraph you're referring to as the total space of a fiber bundle with fiber and base a complex manifold need not be a complex manifold; see here.
The manifolds $S^{2n+1}\times S^{2m+1}$ do admit a complex structure which can be obtained as a quotient of $(\mathbb{C}^{n+1}\setminus\{0\})\times(\mathbb{C}^{m+1}\setminus\{0\})$ by the $\mathbb{C}$-action $t\cdot(x, y) = (e^tx, e^{\alpha t}y)$ where $\alpha$ is a complex number with $\operatorname{Im}(\alpha) \neq 0$. They are called Calabi-Eckmann manifolds.