As it was mentioned in the comments, this is basically writing down the definition.

Since all the $S_i$'s commute with each other, then each component of $\boldsymbol{S}^n$ is of the form $(S_1 \ldots S_1)(S_2 \ldots S_2) \ldots (S_d \ldots S_d)$, where $S_i$ apprears $\alpha _i$ times and $\sum _i \alpha _i =n$. The number of times the sequence $\alpha = (\alpha _1 , \ldots , \alpha _d)$ apprears, is the number of ways you can choose the first option(which is $S_1$) $\alpha_1$ times, the second option (that is $S_2$) $\alpha_2$ times and so on, and this is given by the binomial expansion ${n \choose{\alpha_1, \alpha_2, \ldots , \alpha_d }} = \frac{n!}{\alpha_1!\ldots \alpha_d!}$.

Now by definition, we have: $$ \Vert \boldsymbol{S}^n \Vert^2 = sup \Big\{ \sum _{k=1}^{d^n} \Vert S_1 ^{\alpha _1} S_2^{\alpha_2} \ldots S_d^{\alpha_d} \Vert^2, x\in E, \Vert x \Vert =1 \Big\} $$ which you can see is the same as the left hand side.

As it was mentioned in the comments, this is basically writing down the definition.

Since all the $S_i$'s commute with each other, then each component of $\boldsymbol{S}^n$ is of the form $(S_1 \ldots S_1)(S_2 \ldots S_2) \ldots (S_d \ldots S_d)$, where $S_i$ apprears $\alpha _i$ times and $\sum _i \alpha _i =n$. The number of times the sequence $\alpha = (\alpha _1 , \ldots , \alpha _d)$ apprears, is the number of ways you can choose the first option(which is $S_1$) $\alpha_1$ times, the second option (that is $S_2$) $\alpha_2$ times and so on, and this is given by the binomial expansion ${n \choose{\alpha_1, \alpha_2, \ldots , \alpha_d }} = \frac{n!}{\alpha_1!\ldots \alpha_d!}$.

Hence, we get: $$||S^n||^2=\sup_{||x||=1} \sum_{|\alpha|=n}\frac{n!}{\alpha!}||S^{\alpha}x||^2,$$

because the number of $ S_{i_1}S_{i_2}...S_{i_n}$ in the $d^n$ tuple which will become $S_1^{\alpha_1}...S_d^{\alpha_d}$ is $\frac{n!}{\alpha!}$, where each $i_j\in \{1,2,...,d\}$.