Down to earth, intuition behind a Anabelian group An anabelian group is a group that is “far from being an abelian group” in a precise sense: It is a non-trivial group for which every finite index subgroup has trivial center.

I would like to know what is the motivation behind this definition.

 A: I recommend you this article by Kirsten Wickelgren to supplement the links Najib provides, which establishes the motive for constructing anabelian objects.

In an Anabelian Scheme, the solutions are controlled
  not by the usual algebraic manipulations but rather by
  using the loops on the complex solutions together with a
  Galois group. This difference opens up new possibilities
  for understanding the solutions, which is one reason
  to care about Anabelian Schemes.

A: This is not the context in which the word 'anabelian' arose, so it is not 'the' motivation, but there is at least 'a' motivation for an equivalent definition.  Given an infinite group $G$, one can define the 'virtual centre' or 'FC-centre' to be the set of elements whose centralizer has finite index.  The class of groups defined as 'anabelian' in the question are then the infinite groups with trivial virtual centre.
As an example of why you might want to restrict to this class of groups: if $G$ is an infinite group with trivial virtual centre, there is a well-behaved notion of the group $A$ of virtual automorphisms of $G$.  The group $A$ is then universal for the class of groups $H$ that contain $G$ as a commensurated subgroup (that is, a subgroup such that $|G:G \cap hGh^{-1}|$ is finite for all $h \in H$) with trivial centralizer.  See for instance:
Y. Barnea, M. Ershov and T. Weigel, Abstract commensurators of profinite groups, Trans. AMS 363 (2011), 5381-5417.
