Classically, Grothendieck's motives are only the pure motives, meaning abelian-ish things which capture the (Weil-cohomology-style) $H^i$ of smooth, projective varieties. To see the relationship with motivic cohomology, one should extend the notion of motive so that non-pure (i.e. "mixed") motives are allowed, these mixed motives being abelian-ish things which capture the $H^i$ of arbitrary varieties. The main novelty with mixed motives is that the (conjectural) abelian category of them is not semi-simple -- in fact every mixed motive should be a (generally non-trivial) iterated extension of pure motives, these extensions essentially coming from compactification and resolution of singularities, as in the story of mixed Hodge structures.
Then once one thinks of mixed motives, a natural direction of study (or speculation, as the case may be...) is that of determining all possible extensions (or iterated extensions) between two motives. And that's what motivic cohomology is, essentially: the study of these Ext groups. More formally, every variety $X$ should determine an object $C(X)$ in the bounded derived category of mixed motives, collecting together all the various mixed motives $H^i(X)$, and the $(i,j)^{th}$ motivic cohomology of $X$ is (up to twisting conventions) the abelian group of maps from the unit object to $C(X)$ \ $[i](j)$ (the $j^{th}$ Tate twist of the $i^{th}$ shift of $C(X)$) in the derived category of mixed motives.
Now, there are a few points to make here. The first is that, though the above motivation and definition of motivic cohomology rely on an as-yet-conjectural abelian category of mixed motives, one can, independently of any conjectures, define a triangulated category which, as far as anyone can tell, behaves as if it were the bounded derived category of this conjectural abelian category. The most popular such definition, because of its simplicity and relative workability, is Voevodsky's. So the basic theory and many basic results on motivic cohomology are unconditional.
Another thing to say is that, as always, matters with motives are illuminated by considering realization functors. Let me single out the $\ell$-adic etale realization, since its extension from pure to mixed motives is straightforward (unlike for Hodge structures): any mixed motive, just as any pure motive, yields a finite-dimensional $\ell$-adic vector space with a continuous action of the absolute Galois group of our base field. It then "follows" (in our conjectural framework... or actually follows, without quotation marks, in Voevodsky's framework) that the $(i,j)^{th}$ motivic cohomology of X maps to the abelian group of maps from the unit object to $C^{et}(X)$ \ $[i](j)$ in the bounded derived category of $\ell$-adic Galois representations. But this abelian group of maps is just the classical (continuous) $\ell$-adic etale cohomology $H^i(X(j))$ of the variety $X$, making this latter group the natural target of an $\ell$-adic etale "realization" map from motivic cohomology.
So here comes the third point: note that this is the etale cohomology of $X$ itself, not of the base change from $X$ to its algebraic closure. So this etale cohomology group mixes up arithmetic information and geometric information, and the same is true of motivic cohomology in general. (Think especially of the case $X=pt$: the motivic cohomology of a point admits a generally nontrivial realization map to the $\ell$-adic Galois cohomology of the base field.) For example, it is expected (e.g. by Grothendieck -- see http://www.math.jussieu.fr/~leila/grothendieckcircle/motives.pdf for this and more) that for an abelian variety $A$ over an ``arithmetic'' base field $k$, the most interesting part of the motivic cohomology $H^{(2,1)}(A)$ (again my twists may be off...), by which I mean the direct summand which classifies extensions of $H^1(A)$ by $H^1(G_m)$, should identify with the rationalization of the abelian group of $k$-rational points of the dual abelian variety of $A$, the map being given by associating to such $k$-rational point the mixed motive given as $H^1$ of the total space of the corresponding $G_m$-torsor on $A$. And in this case, the above "realization" map to $\ell$-adic etale cohomology is closely related to the classical Kummer-style map used in the proof of the Mordell-Weil theorem.
So in a nutshell: motivic cohomology is very related to motives, since morally it classifies extensions of motives. But it is of a different nature, since it is an abelian group rather than an object of a more exotic abelian category; and it's also quite different from a human standpoint in that we know how to define it unconditionally. Finally, motivic cohomology realizes to Galois cohomology of a variety itself, rather than to the base change of such a variety to the algebraic closure.
Hope this was helpful in some way.
Best Answer
Two clarifications:
For anabelian geometry, you should ask how much information about a variety is contained in the Galois action on its etale fundamental group.
While it's true that the motivic Galois group is a higher-dimensional analogue of the Galois group, it also should be true that motives are "just" a special kind of Galois representation, i.e. under the Tate conjecture the $\ell$-adic realization functor should give a faithful functor from motives to $\ell$-adic Galois representations, so the category of motives is the category of Galois representations with some restrictions placed on the objects and morphisms.
Of course these restrictions are highly nontrivial. Only for the irreducible Galois representations do we have a good conjectural description of which ones come from motives, via the Fontaine-Mazur conjecture.
So we can see that all 3 of these relate to Galois actions - the first two to Galois actions on fundamental groups, and the last to Galois actions on $\ell$-adic vector spaces. However, Galois actions are used in different ways in the three. Thinking about the three concepts, you might be led to questions like these:
Can we construct Galois representations from the Galois action on the fundamental group of a curve? (this would be the first step in relating motives to anabelian geometry)
Do these Galois representations arise from motives? (this would be the second step)
Are these motives related to the geometry of a curve? (seeking a deeper connection to anabelian geometry)
Can the class of motives arising this way be used to construct or describe the motivic Galois group? (now we bring in Grothendieck-Teichmuller theory)
I think these questions at least touch on the beginning of what Grothendieck was thinking of.
Since Grothendieck, people have heavily studied these questions, primarily in the case of unipotent quotients of the fundamental group, starting with the paper of Deligne on the fundamental group of the projective line minus three points. I think it's fair to say that the answer to all these questions is yes, with the largest caveat for the last question - I believe we can understand certain very special quotients of the motivic Galois group this way, but I don't think anyone has a strategy to construct the whole thing.
The story goes something like this:
Deligne looked at the maximal pro-$\ell$ quotient of the geometric fundamental group of the projective line minus three points. This is naturally an $\ell$-adic analytic group, and has a Lie algebra, which is an $\ell$-adic representation, and admits an action of the Galois group. This is supposed to be the $\ell$-adic realization of a motive, and Deligne worked to find the other realizations, including Hodge theory. This is a mixed motive, not a pure motive, so isn't constructed directly from linearizing algebraic varieties.
All motives generated this way are mixed Tate motives, i.e extensions of powers of the Tate motive (the inverse of the Lefschetz motive), and are everywhere unramified. One can define the Tannakian category of everywhere unramified mixed Tate motives, and the Tannakian fundamental group is known to ask faithfully on the limit of these unipotent completions.