Important Edit: I e-mailed Jacob Lurie, and he said that the statement of condition (*) is incorrect as printed.Here is the correct statement of (*):

For any cofibration $f:A\to B$ and any

trivialfibration $g:X\to Y$ in $C$, the induced morphism:$$\operatorname{Map}(B,X)\to \operatorname{Map}(B,Y)\times_{\operatorname{Map}(A,Y)} \operatorname{Map}(A,X)$$

is a

trivialKan fibration. (Where $\operatorname{Map}$ is the (sSet)-enriched $\operatorname{Hom}$).

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 equivalences 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 morphism:

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

is a Kan fibration. (Where $\operatorname{Map}$ is the (sSet)-enriched $\operatorname{Hom}$).

Lastly, assume that $A\otimes \Delta^n\tilde{\to}A\otimes \Delta^0=A$ is a weak equivalence for any object $A$ in $C$ and any $n\in \mathbf{N}$. (Here, the tensor $A\otimes K$, where $A$ is in $C$ and $K$ is in $sSet$, is the object of $C$ representing the functor $Map(A,-)^K$).

Let $L\subseteq K$ be an inclusion of simplicial sets. Suppose $\sigma:\Delta^n\hookrightarrow K$ 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. Now, the question here is, why is this the case?

The proof I'm reading says that it follows from the line marked (*) above (and the fact that $C$ is left-proper (which follows from the fact that all objects of $C$ are cofibrant)), 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).

Edit: It's probable that a few of the hypotheses are unrelated to the actual question. I included everything because I'm not sure what's important.