I'm attempting to interpret a question, given:
Let $L$ be a first-order language, $\Gamma$ be a set of sentences of $L$, $\forall xFx$ a sentence of $L$, and $c$ a constant not in in $L$.
The intent being to prove that if $\Gamma$ is satisfiable then the union of $\Gamma$ and $\forall xFx \implies F(c)$ is satisfiable.
How am I to interpret the constant $c$ being not in in $L$? Does this mean it is to be considered as anything?
Since $F(c)$ appears to me meaningless if $c$ is not in the language...
$c$ is a new constant symbol added to the original language $\mathcal L$.
In this way, we have a new language $\mathcal L′ = \mathcal L \cup \{ c \}$.
In this new language we have all the previous formulas plus the new formulas using $c$, like: $∀xF(x) → F(c)$.
An interpretation $\mathfrak I$ for the new language $\mathcal L′$ will be an interpretation that gives meaning also to the symbol $c$, i.e. that assigns to $c$ an object of the domain of the interpretation (called $c^{\mathfrak I}$).
What does it mean to prove that $∀xF(x) → F(c)$ is satisfiable?
It means to consider an interpretation $\mathfrak I$ suitable for the language $\mathcal L'$ (and thus an interpretation where the constant $c$ has meaning) and show that the formula is satisfied in $\mathfrak I$.