Using the étale fundamental group one can construct an injective group homomorphism
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \hookrightarrow \operatorname{Out}(\widehat{F_2})$
which is canonical in the sense that there are no choices involved in its construction (once the algebraic closure is fixed), $\operatorname{Out}$ refers to the outer automorphism group and $\widehat{F_2}$ to the profinite completion of the free group on two letters.
As for your example of a "WOOOOW" moment, this statement no longer contains the étale fundamental group in its statement, even though it's vital for the construction.
The fact that the absolute Galois group of the rationals is canonically a subgroup of the outer automorphisms of a (profinite-) free group is completely non-obvious. (Try to prove it from scratch...)
One can try to determine the image of this map. This leads to the profinite Grothendieck-Teichmueller group, which sometimes is conjectured to be isomorphic to $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.
The étale fundamental group enters because one considers étale coverings of the projective line minus three points (i.e. the affine line minus two points). Over an algebraically closed field of characteristic zero, this has the étale fundamental group $\widehat{F_2}$ (think of the loops around two of the removed points as its generators). But now study the projective line minus three points over the rationals instead of their algebraic closure, this makes the Galois group of the rationals enter.
The projective line minus three points enters because by a construction due to Belyi, every algebraic curve which can be defined over a finite extension of the rationals can be realized as an étale covering of the projective line minus three points (this is an if and only if). The idea is to modify a map to the projective line so that its ramification gets concentrated in only three points.
You will find the rest of the story for example in surveys by Leila Schneps on the Grothendieck-Teichmueller theory.
