How to interpret a formula with free variables ?
In mathematics, we usually have two kind of "equations" :
The first one is an identity and it is clearly implicitly universally quantified; i.e. it must be read as :
$\forall x [(x+1)(x-1)=x^2-1]$.
If we consider for simplicity the interpretation based on the domain $\mathbb N$ of natural numbers, and we instantiate it with a number $n$ whatever, we always get a true formula : $(n+1)(n-1)=n^2-1$, i.e. the "matrix" (the sub-formula without the quantifier) is always satisfied.
This is simply a consequence of the logical axiom (or law) :
$\forall x \varphi \rightarrow \varphi_t^x$, where $t$ is substitutable for $x$ in $\varphi$.
Formula $x^2-2x+1=0$, instead, must not be read as universally quantified, because it is simply false that every number $n \in \mathbb N$ satisfy it.
It must be read as :
$\exists x (x^2-2x+1=0)$.
How to interpret $x^2-2x+1=0$ with $x$ free ? To do this, we have to assign a "temporary" denotation to the free variable: this can be done in more than one way (all more or less equivalent).
We can, for example, use a variable assignement function $s : Var \to D$ where $Var$ is the set of variables of the language and $D$ is the domain of the interpretation : in our example $\mathbb N$.
Thus, if we consider the function $s$ that assigns to $x$ the number $0$ we have that :
is clearly false, because : $0-2 \times 0 +1 = 1 \ne 0$. We say that $s$ does not satisfy the formula.
If we consider instead the function $s'$ that assigns to $x$ the number $1$ we have that :
is true, because : $1-2+1 = 0$, and we say that $s'$ satisfy the formula (by the way, this shows that the formula $\exists x (x^2-2x+1=0)$ is true in $\mathbb N$, due to the fact that we have found a variable assignment that satisfy its "matrix").
The above argument is also the explanation of the assertion that "free variables and constants play the same role"; in order to show the satisfiability of a formula with a free variable, we treat the variable as a "temporary" name for an object in the domain of the interpretation (we assign to $x$ a denotation through the variable assignment function $s$).