Let $C$ be a simplicially-enriched category with a model structure, not necessarily compatible with the simplicial enrichment.

Suppose that all objects of $C$ are cofibrant, that $C$ is tensored and cotensored over $SSet$, and that the class of weak equivalence of $C$ is closed under filtered colimits.

(*)Suppose further that for any cofibration $f:A\to B$ and any fibration $g:X\to Y$ in $C$, the induced morphisms:

$$Map(B,X)\to Map(B,Y)\times_{Map(A,Y)} Map(A,X)$$

is a Kan fibration.

Lastly, assume that $A\otimes \Delta^n\tilde{\to}A\otimes \Delta^0\cong A$ for any object $A$ in $C$.

Let $L\subseteq K$ be an inclusion of simplicial sets. Suppose $\sigma:\Delta^n\to K$ is a nondegenerate simplex of $K$ with all of its faces living in $L$. That is, we can factor the map $\partial\sigma:=\sigma|_{\partial\Delta^n}$ through the inclusion $L\subseteq K$ (in fact, we will assume that the target of this map actually is $L$).

Then for any object $D$ in $C$, the pushout $$D\otimes \Delta^n\coprod_{D\otimes\partial\Delta^n} D\otimes L\cong D\otimes (\Delta^n\coprod_{\partial\Delta^n}L)$$ is a homotopy pushout. Since the model structure is left-proper, it's enough to show that one of the induced maps $D\otimes\partial\Delta^n\to D\otimes\Delta^n$ or $D\otimes\partial\Delta^n\to D\otimes L$ in the cocartesian square is a cofibration.

The proof I'm reading says that it follows from the line marked (*) above, but it's not clear to me how to apply that hypothesis.

That is, how does the line marked (*) imply anything relevant?

If you'd like to look up the original source, it is Higher Topos Theory Proposition A.3.1.7 (in the appendix).