Over a field of characteristic coprime to $n$, the classifying space ${\rm B}_{\rm et}\mu_n$ is $\mathbb{A}^1$-local. In this situation, any $\mu_n$-covering $\widetilde{X}\to X$ is an $\mathbb{A}^1$-covering space and provides a classifying morphism $X\to {\rm B}_{\rm et}\mu_n$. From the Kummer sequence $\mu_n\to \mathbb{G}_{\rm m}\to \mathbb{G}_{\rm m}$ we get a fiber sequence ${\rm B}_{\rm et}\mu_n\to {\rm B}\mathbb{G}_{\rm m}\to {\rm B}\mathbb{G}_{\rm m}$. This implies that the sheaf of $\mathbb{A}^1$-connected components is $\mathbb{G}_{\rm m}/\mathbb{G}_{\rm m}^n$ (the Nisnevich sheaf quotient) which we can alternatively write as the Nisnevich sheafification of ${\rm H}^1_{\rm et}(-,\mu_n)$. (This is in the Morel-Voevodsky paper.)
Now turn to $\mathbb{A}^m\setminus\{0\}$. The $\mu_n$-covering $\mathbb{A}^m\setminus\{0\}\to\mathbb{A}^m\setminus\{0\}/\mu_n$ is classified by a map $\mathbb{A}^m\setminus\{0\}/\mu_n\to {\rm B}_{\rm et}\mu_n$ and the more precise claim is that the induced map on $\pi_0^{\mathbb{A}^1}$ is an isomorphism of Nisnevich sheaves.
For surjectivity, we consider an embedding $\mathbb{G}_{\rm m}\to\mathbb{A}^m\setminus\{0\}$ as $i$-th component. This is a $\mu_n$-equivariant map if we take the componentwise action on $\mathbb{A}^m\setminus\{0\}$. Therefore, we get an induced map on quotient schemes $\mathbb{G}_{\rm m}\to \mathbb{A}^m\setminus\{0\}/\mu_n$. On sheaves of $\mathbb{A}^1$-connected components we get induced maps
$$
\mathbb{G}_{\rm m}\to \pi_0^{\mathbb{A}^1}\mathbb{A}^m\setminus\{0\}/\mu_n\to \mathbb{G}_{\rm m}/\mathbb{G}_{\rm m}^n
$$
such that the composition is the natural quotient map. This proves the surjectivity.
We check injectivity locally in the Nisnevich topology. For that, let $R$ be a henselian local ring. A morphism ${\rm Spec}R\to \mathbb{A}^m\setminus\{0\}/\mu_n$ is represented by a tuple $(x_1,\dots,x_m)$ with $x_i\in R$ such that at least one of the $x_i$ is not in the maximal ideal of $R$. Two such tuples $(x_1,\dots,x_m)$ and $(y_1,\dots,y_m)$ give the same morphism if the tuples $(x_1^n,\dots,x_m^n)$ and $(y_1^n,\dots,y_m^n)$ agree. Now we want to show that any map ${\rm Spec} R\to \mathbb{A}^m\setminus\{0\}/\mu_n$ (corresponding to a tuple $(x_1,\dots,x_m)$) factors through a coordinate embedding $\mathbb{G}_{\rm m}\to\mathbb{A}^m\setminus\{0\}/\mu_n$ used before. One entry $x_i$ is not contained in the maximal ideal, assume it's the first. Then $(x_1,t\cdot x_2,\dots,t\cdot x_m)$ provides a naive $\mathbb{A}^1$-homotopy from the given tuple to $(x_1,0,\dots,0)$. So up to $\mathbb{A}^1$-homotopy, any map ${\rm Spec} R\to \mathbb{A}^m\setminus\{0\}/\mu_n$ factors through a coordinate embedding $\mathbb{G}_{\rm m}\to \mathbb{A}^m\setminus\{0\}$. If such a homotopy class maps to the base point of $\mathbb{G}_{\rm m}/\mathbb{G}_{\rm m}^n$ then by the identifications above, it must lift through the $\mu_n$-covering $\mathbb{G}_{\rm m}\xrightarrow{n} \mathbb{G}_{\rm m}$. But since we have a morphism from the covering $\mathbb{G}_{\rm m}\to \mathbb{G}_{\rm m}$ to the covering $\mathbb{A}^m\setminus\{0\}\to \mathbb{A}^m\setminus\{0\}/\mu_n$, the map ${\rm Spec}R\to \mathbb{A}^m\setminus\{0\}/\mu_n$ lifts (up to $\mathbb{A}^1$-homotopy) through a map ${\rm Spec}R\to \mathbb{A}^m\setminus\{0\}$. But the target there is $\mathbb{A}^1$-connected, meaning that the original map ${\rm Spec} R\to \mathbb{A}^m\setminus\{0\}/\mu_n$ must be null-homotopic, showing injectivity.
