We want to prove the model existence lemma: $\mathcal{\varGamma}$ is a consistent set of $\mathcal{L}$-sentences $\Leftrightarrow$ $\mathcal{\varGamma}$ has a model.
In the Henkin-style proof, we make a model $\mathcal{M}$ for $\mathcal{\varGamma}$ [By making a model for the maximal and Henkin $\mathcal{L_{\omega}}$-theory $T'$ that is extended over the $\mathcal{L}$-theory $T=${$A:\Gamma\vdash A$}]
$*$Hint: $\mathcal{L_{\omega}}$ is the extension of $\mathcal{L}$ by adding constants corresponding to the existensial formulas in $\mathcal{L}$ [Infinite union over recursively defined extentions of $\mathcal{L}$].
The process is well-known and I won't mention it in detail. Okay, surely it is a model for $\mathcal{\varGamma}$, But $\mathcal{M}$ is a $\mathcal{L_{\omega}}$-model.
Now, the question is: Weren't we supposed to find a $\mathcal{L}$-model for $\mathcal{\varGamma}$? If yes, then what's worthy about the $\mathcal{L_{\omega}}$-model $\mathcal{M}$? And if no, please clarify me about what's going on!
Thanks
Let $M$ be an $L_\omega$-structure which is a model of $\Gamma$, and let $M_0$ be the $L$-structure whose underlying set is the same as the underlying set of $M$, and such that the interpretations of the non-logical symbols of $L$ are the same as their interpretations in $M$.
All sentences of $\Gamma$ are true in $M_0$, so the $L$-structure $M_0$ is a model of $\Gamma$.