The proof of triviality is a step in the famous Neukirch-Uchida theorem of anabelian geometry, which says a number field is characterized by its absolute Galois group, even functorially, in an appropriate sense. The key elementary fact is the following:
Let $k$ be a number field, $K$ an algebraic closure, and $G=Gal(K/k)$. Let $P_1$ and $P_2$ be two distinct primes of $K$ with corresponding decomposition subgroups $G(P_i)\subset G$. Then
$G(P_1)\cap G(P_2)=1.$
Once this is stated for you, it's essentially an exercise to prove.
Determining the center of $G$ becomes then completely straightforward: Suppose $g$ commutes with everything. Then for any prime $P$, $G(gP)=gG(P)g^{-1}=G(P)$. So $g$ must fix every prime, implying that it's trivial.
I think this is spelled out in the book Cohomology of Number Fields, by Neukirch, Schmidt, and Wingberg. Unfortunately, I left my copy on the plane last year.
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
This article from Le Monde (in French) and this blog (in English) are recent and seem to accurately sum up the state of affairs: In April 2012 Jean Malgoire (see the video interview in the blog) donated the manuscript of "La longue Marche" (700 handwritten pages written by Alexander Grothendieck in just 20 days in 1981) to the library of Montpellier University. They have announced on 17 June 2015 that they will digitize the manuscript (and other documents from A.G.'s estate) for purposes of conservation (apparently the text is fading). Distribution can only happen after permission from the heirs. No indication that this will happen anytime soon.
An earlier MSE posting indicated that one can write to prof. Malgoire to obtain transcriptions of "La longue Marche". The relevant web page was removed in 2007 (it survives in this archive), and I presume the present legal situation regarding the estate of A.G. would prevent any such informal dissemination (beyond what is already available at the Grothendieck circle).