For me, it's one of those questions that would not be so interesting if the answer is Yes but which would probably be very interesting if the answer is No. If not all groups are Galois groups over Q, then there is probably some structure that can be regarded as an obstruction, and then this structure would probably be essential to know about. For instance, not all groups are Galois groups over local fields -- they have to be solvable. This is by basic properties of the higher ramification filtration, which is, surprise, essential to know about if you want to understand local fields. So you could say it's an approach to finding deeper structure in the absolute Galois group. Why not just do that directly? The problem with directly looking for structure is that it's not a yes/no question, and so sometimes you lose track of what exactly you're doing (although in new and fertile subjects often you don't). So the inverse Galois problem has the advantage of being a yes/no question and the advantage that things would be really interesting if the answer is No. Unfortunately, I think the answer is expected to be Yes, though correct me if I'm wrong.
The basic Grothendieck's assumptions means we are dealing with an connected atomic site $\mathcal{C}$ with a point, whose inverse image is the fiber functor $F: \mathcal{C} \to \mathcal{S}et$:
(i) Every arrow $X \to Y$ in $\mathcal{C}$ is an strict epimorphism.
(ii) For every $X \in \mathcal{C}$ $F(X) \neq \emptyset$.
(iii) $F$ preseves strict epimorphisms.
(iv) The diagram of $F$, $\Gamma_F$ is a cofiltered category.
Let $G = Aut(F)$ be the localic group of automorphisms of $F$.
Let $F: \widetilde{\mathcal{C}} \to \mathcal{S}et$ the pointed atomic topos of sheaves for the canonical topology on $\mathcal{C}$. We can assume that $\mathcal{C}$ are the connected objects of $\widetilde{\mathcal{C}}$.
(i) means that the objects are connected, (ii) means that the topos is connected, (iii) that $F$ is continous, and (iv) that it is flat.
By considering stonger finite limit preserving conditions (iv) on $F$ (corresponding to stronger cofiltering conditions on $\Gamma_F$) we obtain different Grothendieck-Galois situations (for details and full proofs see [1]):
S1) F preserves all inverse limits in $\widetilde{\mathcal{C}}$ of objets in $\mathcal{C}$, that is $F$ is essential. In this case $\Gamma_F$ has an initial object $(a,A)$ (we have a "universal covering"), $F$ is representable, $a: [A, -] \cong F$, and $G = Aut(A)^{op}$ is a discrete group.
S2) F preserves arbritrary products in $\widetilde{\mathcal{C}}$ of a same $X \in \mathcal{C}$ (we introduce the name "proessential for such a point [1]). In this case there exists galois closures (which is a cofiltering-type property of $\Gamma_F)$, and $G$ is a prodiscrete localic group, inverse limit in the category of localic groups of the discrete groups $Aut(A)^{op}$, $A$ running over all the galois objects in $\mathcal{C}$.
S2-finite) F takes values on finite sets. This is the original situation in SGA1. In this case the condition "F preserves finite products in $\widetilde{\mathcal{C}}$ of a same $X \in \mathcal{C}$ holds automatically by condition (iv) ($F$ preserves finite limits), thus there exists galois closures, the groups
$Aut(A)^{op}$ are finite, and $G$ is a profinite group, inverse limit in the category of topological groups of the finite groups $Aut(A)^{op}$.
NOTE. The projections of a inverse limit of finite groups are surjective. This is a key property. The projection of a inverse limit of groups are not necessarily surjective, but if the limit is taken in the category of localic groups, they are indeed surjective (proved by Joyal-Tierney). This is the reason we have to take a localic group in 2). Grothendieck follows an equivalent approach in SGA4 by taking the limit in the category of Pro-groups.
S3) No condition on $F$ other than preservation of finite limits (iv). This is the case of a general pointed atomic topos. The development of this case we call "Localic galois theory" see [2], its fundamental theorem first proved by Joyal-Tierney.
[1] "On the representation theory of Galois and Atomic topoi", JPAA 186 (2004)
[2] "Localic galois theory", Advances in mathematics", 175/1 (2003).
Best Answer
(I only just saw this one year on!) I find your question very strange. Grothendieck gives a simple categorical formulation of a situation that encompasses the three main examples of Galois theoretic machines. That means he shows what makes things really tick... isn't that good enough for you! He does this with the clearly stated aim of developing a fundamental group for schemes, and the theory gives that and a lot more. If you go to the slightly wider results on the fundamental groupoid of categories of locally finite sheaves, that is a first step towards his Pursuing Stacks, the letters to Larry Breen, and enroute for his Longue Marche.
In another direction it provides a first step towards the Joyal-Tierney theory of locales etc. and their relation with toposes. It provides a background for all of Jacob Lurie's work on higher toposes, and I could go on with fundamental groups of toposes, homotopy theory of toposes. SGA1 is the key for understanding a large part of modern mathematics.
Grothendieck's methodology was always to seek the clarity that came from abstraction and generalisation. His aim was not only to solve problems (say in algebraic geometry) but to understand as fully as possible their solution and why they worked.