I'm not sure I understand what you precisely want to know, so I'll try to clarify a few things from the topos theoretic point of view, which I hope will answer your questions:
Relation to Galois theory: Classical Galois theory tells you that the category of étale extention of your fields $k$ is equivalent (contravariantly) to the category of finite sets endowed with a continuous action of the galois group (well technically it tells you that transitive action corresponds to fields extention, but étale extention are finite product of fields extention and general action are finite coproduct of transitive action).
When you look at the category of sheaves on the category of finite action with the natural topology (covering are surjection of finite action) you gets rather trivially the topos of all continuous action of the galois group (basically because they are justs non-finite coproducts of the finite action).
And conversely, if you know that the étale topos is the topos of sets with a continuous $G$ action, as étale extention corresponds to representable sheaves one easily deduce (with more or less work) the classical version of Galois theory.
Topos theoretic Galois theory: If what you had in mind is the formalism of Grothendieck of looking at the automorphism of a fiber functor, this is also something we do a lot in topos theory and it is indeed closely related to this representation theorem for the étale topos.
In fact a relatively direct extension of Grothendieck Galois theory can be phrased as follow: if you have a connected atomic topos $P$ with a point $p$ then taking the fiber at $p$ induce an equivalence between $T$ and the category of sets endowed with a continuous action of the group of automorphism of $p$. If one wants the theorem to be true in full generality one need to use a 'localic group' but for the toposes which comes from algebraic geometry it is not a problem as those groups are always compact, in fact the original formulation of Grothendieck of his galois theory corresponds precisely to the case where the group is compact. See for example this paper of E.Dubuc for a presentation of these ideas.
The étale topos and the étale fundamental group: For the étale topos of a scheme, things are a little bit more complicated, there is still a close relationship between the étale topos and the classifying topos of the étale fundamental group (I mean the topos of set with a continuous action of the étale fundamental group), but they are not isomorphic at all.
The classifying topos of the étale fundamental group is closely related to the category of locally constant objects of the étale topos. This is completely similar to the fact that when you have a topological space $X$ the category of locally constant sheaves over $X$ is equivalent to the category of set endowed with an action of the $\pi_1$ (and if you look to space that are not semi-locally simply connected then the $\pi_1$ gets a topology and you have to look at continuous action as for the étale fundamental group and you do not get exactly the locally constant sheaves but something related).
In fact there is a general theory of the $\pi_1$ of toposes, where you essentially have that for a general (grothendieck) topos you can prove an equivalence between "covering projection" (which are the same as locally constant object in good cases or if you are only interested in finite covering projection) and sets with continuous action of the $\pi_1$. (also in the most general version, you will need a groupoids and the groups have to be localic). I believe there are several variant of this which are not all fully general and not all clearly equivalent, but this paper also by Dubuc is probably one of the best place to look. the terminology of "covering projection" that I used above is introduced in this paper.
It is important to keep in mind that for a general scheme this only corresponds to a small part of the étale topos: the ``locally constant objects'' (at least for the finite ones), it justs happen that for a fields "because it is only a point" (informally) everythings is always locally constant. In general the étale fundamental groups is just the $\pi_1$ of the étal topos, it does not see the full space.
General representation theorem for the étale topos: But this is not the end of our geometric understanding of the étale topos. In a hugely important (and sadly hard to found) paper called "an extention of the galois theory of Grothendieck", A.Joyal and M.Tierney proved that this idea of Grothendieck Galois theory can be extended (and coupled with the notion of descent theory in algebra) to obtain a very general representation theorem for toposes: any Grothendieck topos is the category of equivariant sheaves of sets on a localic groupoids. They moreover have a rather explicit construction of the groupoids (but it needs a fair amount of expertise on the subject to be used).
There is a variant of this construction due to C.Butz and I.Moerdijk (Representing topoi by topological groupoids) which applies to topos with enough points (which is the case of the étale topos by a famous theorem of Deligne) and produce a topological groupoid instead of a localic groupoid, but most importantly is a lot more explicit, at least if you have a good understanding of the points of our topos and more generally of what it classifies (which in the case of the étale topos is well known: for affine scheme they are the strict localization of your ring).
If you go through the construction of Butz & Moerdijk for the étale topos you get that it can be represented by a topological groupoid whose objects corresponds to a large enough set of strict localization of your ring with a certain topology a bit similar to a Zariski topology, with morphism being essentially all the local galois action on those strict localization. The full description of the groupoid is a little bit involved, and I do not know if it has been worked out in details for the étale topos of a scheme (I'm not very familiar with the literature on the subject, maybe someone else can add a reference here, or say if that does not exists ?)
