I have started studying free groups but cannot get motivation behind it. I know that a free group $F_A$ over the set $A$ consisting of all combinations or expressions that can be built from the members of $A$.
But why would one want to do such a construction? What is the motivation behind it
Well, off the top of my head, there are two big things that free groups do for us.
First of all, they give us a nice generalization of the integers under addition--also known as the free group generated by one element. (One could also argue that the trivial group is the free group generated by zero elements.)
Second of all, every group is the quotient of a free group! Have you ever seen something like the following before?
$$D_n=\left\langle x,y\mid x^n,y^2,xyxy\right\rangle$$
This is shorthand that says $D_n$ can be obtained by starting with the free group $F$ generated by two elements--$x$ and $y,$ say--letting $N$ be the smallest normal subgroup of $F$ containing each of $x^n,y^2,$ and $xyxy,$ and then forming the quotient group $F/N.$