The following theorem is given:
If each finite subset of a set $\Sigma$ of first-order formulas has a non-empty model, then $\Sigma$ has a non-empty model.
Then, how can one deduce the following theorem?:
If each finite subset of a set $\Sigma$ of first-order formulas has a (not necessarily non-empty) model, then $\Sigma$ has a (not necessarily non-empty) model.
If every finite subset of $\Sigma$ has a nonempty model, you're done by the given theorem. So you only have to consider the case that some finite subset $F\subseteq\Sigma$ has only the empty structure as a model. It follows that if $G\supseteq F$, $G$ can only have the empty structure as a model. Since every finite subset of $\Sigma$ has a model, the empty structure must be a model of $F\cup\{\varphi\}$ for all $\varphi\in\Sigma$. That is, the empty structure is a model of every sentence in $\Sigma$.