I recommend reading about "root stacks" as studied by Charles Cadman. Given an invertible sheaf, e.g., $\mathcal{O}_X(mK_X)$, and given an integer $m$, there is an Artin stack $\mathcal{R}\to X$ parameterizing $m^{\text{th}}$ roots of the pullback of this invertible sheaf. This stack is a $\mu_m$-gerbe over $S$, hence it is Deligne-Mumford in characteristic $0$. Over the smooth locus $U$ of $X$, there is a section $s_U:U \to \mathcal{R}$, since $\omega_U$ is an $m^{\text{th}}$ root of $\mathcal{O}_X(mK_X)|_U$. The claim is that there is a unique triple $(\mathcal{X}_m,s,i_m)$ of a normal, separated Artin stack $\mathcal{X}_m$, a finite morphism $s:\mathcal{X}_m \to \mathcal{R}$, and a morphism over $\mathcal{R}$, $i_m:U\to \mathcal{X}_m$ such that $i_m$ is representable by dense open immersions. In particular, this means that $s$ is the "integral closure of $\mathcal{R}$ in the field of fractions of $U$". Since that construction is compatible with smooth base change, you can construct the triple étale locally over $X$, which is what Kawamata describes. Finally, you can check that the natural induced morphism $\mathcal{X}_m \to \mathcal{X}_{mn}$ is an isomorphism. So the construction (locally) only depends on the "local index" of $K_X$.