This is really about how you evaluate the truth value.
$\exists x\varphi(t)$ is true if and only if there exists some $x$ for which $\varphi(x)$ is true. Conversely it is false if and only if for all $x$ (in a given model, of course) $\varphi(x)$ is false.
The inner quantification is mostly to "confuse" your intuition and since the truth value of $\forall x\forall y(P(x,y))$ is not dependent of the truth value of the outer quantification it is easier to change the variables, even informally before writing the actual proof.
The claim itself just says that there is a pair $(x,y)$ such that if $P(x,y)$ then $P$ is all the ordered pairs of the universe.
We can prove the validity of this formula from an external point of view, and we do that semantically (that is we do not try to write a proof, but rather show that is holds in every model), for brevity denote our formula $\varphi$.
Let $M$ be an arbitrary model of our language (where $P$ is a binary relation).
If $M\models\forall x\forall y(P(x,y))$ then $M\models\varphi$ (can you see why?)
If $M\models\lnot(\forall x\forall y(P(x,y))$ then for some $a,b\in|M|$ we have $\lnot P(a,b)$. In particular for this pair that $M\models P(a,b)\rightarrow\forall x\forall y(P(x,y))$, so we have $M\models\varphi$.
If needed, this should be made rigorously using the $\operatorname{val}$ function. I strongly recommend on working the details yourself and following closely after the definitions of $\operatorname{val}_M(\exists x\varphi,g)$ (and similarly $\forall x\varphi$).
Consider the informal argument 'Everyone loves pizza. Anyone who loves pizza loves ice-cream. So everyone loves ice-cream.' Why is that valid?
Roughly: Pick someone, whoever you like. Then, s/he loves pizza. And so s/he loves ice-cream. But sh/e was arbitrarily chosen. So everyone loves ice-cream.
This informal argument can be spelt out with plodding laboriousness with explicit commentary like this:
Everyone loves pizza. (That's given)
Anyone who loves pizza loves ice-cream. (That's given too)
Take some arbitrary person, call her Alice. Then Alice loves pizza (From 1)
If Alice loves pizza, she loves ice-cream (From 2)
Alice loves ice-cream (From 3, 4, by modus ponens)
But Alice was an arbitrary representative person, so what applies to her applies to anyone: so everyone loves ice-cream.
Now consider the formal analogue of this proof (using $F$ for 'loves pizza" etc.)
$\forall xFx\quad\quad\quad\quad$ Premiss
$\forall x(Fx \to Gx)\quad$ Premiss
$Fa\quad\quad\quad\quad\quad$ From 1., Univ. Instantiation
$(Fa \to Ga)\quad\quad$ From 2., Univ. Instantiation
$Ga\quad\quad\quad\quad\quad$ From 3, 4 by MP
$\forall xGx\quad\quad\quad\quad$ From 5, Univ. Generalization
(UG on $a$ is legimitate as $a$ appears in no assumption on which (5) depends, so can be thought of as indicating an arbitrary representative member of the domain.)
So note, that we need here some symbol playing the role of $a$, not a true constant with a fixed interpretation, not a bound variable, but (as common jargon has it) a parameter which stands in, in some sense, for an arbitrary element of the domaain.
Now, in some syntaxes, variables $x$, $y$ etc. only ever appear bound, as part of quantified sentences. And parameters are typographically quite distinct, $a$ and $b$, etc. This is the usage in Gentzen, for example.
But another tradition (probably more common, but not for that reason to be preferred), we are typographically economical, and re-cycle the same letters as both true variables and as parameters. In other words, we allow the same letters to appear both as "bound" variables and as "free" variables. Then the formal proof will look like this:
$\forall xFx\quad\quad\quad\quad$ Premiss
$\forall x(Fx \to Gx)\quad$ Premiss
$Fx\quad\quad\quad\quad\quad$ From 1., Univ. Instantiation
$(Fx \to Gx)\quad\quad$ From 2., Univ. Instantiation
$Gx\quad\quad\quad\quad\quad$ From 3, 4 by MP
$\forall xGx\quad\quad\quad\quad$ From 5, Univ. Generalization
This is a superficial difference, however: the role of the free (unbound) variable is as a parameter. And we've seen, even in informal arguments, we use expressions like "s/he" or even "Alice" as parameters. Looked at in that light, there should be no mystery about why we need parameters in a formal logic too. And in one syntax, unbound variables play the role of parameters.
Best Answer
If we define the semantics for formulas with free variables, like in Enderton's textbook, we have that a formula $\varphi$ is satisfied in a structure $\mathfrak A$ with a variable assignment $s$, in symbols:
Thus, the formula $x=0$ is satisfiable because the structure $\mathbb N$ with the assignment $s(x)=0$ will do.
A formula $\varphi$ is true in a structure $\mathfrak A$ if it is satisfied with every variable assignment.
A formula is valid iff for every structure $\mathfrak A$ and every assignment $s$, $\mathfrak A$ satisfy $\varphi$ with $s$.