The dlt assumption on $\mathscr{X}$ and the toricness of $Z$ ensure that $Z$ is smooth, and that $(Z,\Delta_{Z})$ is an snc pair. Indeed, the singular locus of $Z$ is a union of torus invariant subvarieties (see [CLS11, Proposition 11.1.2]), hence the generic point of a component of the singular locus is the generic point of a stratum of $\Delta_{Z}$. However, $(Z,\Delta_{Z})$ is a dlt pair, thus snc at the generic point of each stratum of $\Delta_{Z}$.