$\require{mathtools}$
The reason this works is that the inclusion map $i:\Delta[n]\times\Lambda^k[m]\hookrightarrow{}\Delta[n]\times\Delta[m]$ can be viewed as a composition. The maps in this composition are all monomorphisms of the form $i_{\ell}:X\hookrightarrow{}X\cup_{S}\Delta[m+n]$; that is, they adjoin one of the missing simplices, which all happen to be of dimension $m+n$. In particular, this is a pushout of a (generalized) horn inclusion, so it has the left lifting property with respect to Kan complexes. A detailed account of this decomposition can be found in, e.g., the appendix to Rezk's notes on quasicategories, but it's a good exercise to work it out yourself.