For a $S$-Scheme $X$, and $n$ an integer invertible in $S$, denote by $\mu_n$ the sheaf fitting in the following exact sequence:\begin{align} 0 \longrightarrow \mu_n \longrightarrow \mathcal{O}_{X_{et}}^* \overset{n}{\longrightarrow} \mathcal{O}_{X_{et}}^* \longrightarrow 0 \end{align}
I understand that this sheaf is supposedly representing the $n$-th roots of unity of the sheaf $\mathcal{O}_X$, as an abelian group, this is a locally constant sheaf, and in the "nice cases" (like when $S$ is the spectrum of an algebraically closed field), this sheaf is just the constant sheaf associated to the group $\mathbb{Z}/n\mathbb{Z}$.
To define traces in étale cohomology, we first define $\mathbb{Z}/n\mathbb{Z}(d) = \mu_n^{\otimes d}$ for $d \geq 0$ and its dual for $d \leq 0$, and given any sheaf $\mathscr{F}$ on $X$, we define $\mathscr{F}(d) = \mathscr{F} \otimes\mathbb{Z}/n\mathbb{Z}(d)$.
I can work with these definitions, but I have absolutely no idea what they represent, neither how I can interpret or picture the operation "tensoring by $\mu_n(d)$".
How should I understand these operations and object? and what is the intuition behind them?