Firstly a few preliminares.
Definition: Let $G$ denote a compact Lie group acting on a space $X$, its homotopy orbit space $X_{hG}$ is defined as the quotient of the product space $EG × X$ by the diagonal $G$–action, where $EG$ is the universal $G$–space.
$T$ is the maximal torus (product of $n$ factors of spheres) of $U(n)$, $NT$ is the normalizer, and $S_n$ the symmetric group on $n$ letter, which is the the Weyl group $NT/T$.
We know that $\mathbb{C}P^d$ is the quotient of $(S^{2d+1})^n$ by the free action of the maximal torus $T$. Now, since $T$ is a normal subgroup of $NT$, it also acts freely on $(S^{2d+1})^n_{hNT}$. If we first take the quotient of the free action of $T$ and then the residual action of $NT/T = S_n$, we get $(\mathbb{C}P^d)^n_{hS_n}$.
From the comments above, I kind of see why that is true. However, I'm not satisfied with my "hand-waving proof" and I'm getting confused when writing it down properly. Because I'm trying to apply the similar idea to another situation, I'm looking for a detailed proof. Any comment with more details would be helpful. Many thanks!
For any compact Lie group $G$ and $G$-space $X$, if $N$ is a normal subgroup whose action on $X$ is free, we have $X_{hG} \simeq (X/N)_{hG/N}$. To see why, note that $X_{hN}$ with the residual $G/N$-action is modeled by $(EG\times X)/N$ and the map $(EG\times X)/N\to X/N$ induced by collapsing $EG$ to a point is both $G/N$-equivariant and a homotopy equivalence, since the action of $N$ is free. Thus, we obtain a homotopy equivalence after taking the $G/N$-homotopy orbits: $X_{hG}\simeq (X_{hN})_{hG/N}\simeq (X/N)_{hG/N}$.