This follows from the fact that $\mathbb{E}[A^\dagger A]=d I$ (with $A^\dagger$ the conjugate transpose of $A$ and $I$ the $d\times d$ identity matrix). Hence 
$$\operatorname{tr}\mathbb{E}[A_1A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger A_1^\dagger]$$
$$=\operatorname{tr}\mathbb{E}[A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger]\mathbb{E}[ A_1^\dagger A_1]$$
$$=\operatorname{tr}d\mathbb{E}[A_2\cdots A_nA_n^\dagger\cdots A_2^\dagger]$$
$$=\operatorname{tr}d\mathbb{E}[A_3\cdots A_nA_n^\dagger\cdots A_3^\dagger]\mathbb{E}[ A_2^\dagger A_2]$$
$$=\operatorname{tr}d^2\mathbb{E}[A_3\cdots A_nA_n^\dagger\cdots A_3^\dagger]$$
$$=\operatorname{tr}d^{n-1}\mathbb{E}[A_nA_n^\dagger]=\operatorname{tr}d^n I=d^{n+1}.$$