I was wondering if someone could give me an intuitive interpretation of what we have done after abelianizing a group.
I know what formal definition is: once we have our group $G$ given, we take a quotient by the commutator subgroup $[G,G]$, where $[G,G]$ is the unique smallest normal subgroup $N$ such that $G/N$ is abelian.
But who guarantees that this is abelian?
Also I was wondering why is this so helpful if in fact we get weaker (poorer) group than the first one.
Let's say I have a nonabelian group $G$, and I want to "make it abelian". Intuitively, that means that whenever $g,h\in G$, I want $gh$ to be the same as $hg$.
Note that one particular case is that I'll want every element of the form $ghg^{-1}h^{-1}$ to be the identity. Interestingly, this is all that I need to do! If $ghg^{-1}h^{-1}=e$ for all $g, h\in G$, then $G$ is abelian: $$ghg^{-1}h^{-1}=e\implies ghg^{-1}=h\implies gh=hg.$$ So this tells us:
In order to "make $G$ abelian," it's enough to "make every commutator trivial." And, conversely, if I "make $G$ abelian" I will "make every commutator trivial" - so these are the same task.
Now this suggests a thing to do: look at the quotient group gotten by modding out by the commutator subgroup! Intuitively, this should be the group which is the "abelian version" of $G$. Of course, this hinges on the commutator subgroup being normal . . .
. . . which it is! So that's nice.
Now we want to check that everything worked properly. Let's say $a, b\in G/Comm(G)$. Pick representatives $g, h$ of $a, b$ respectively (remember that $a, b$ are equivalence classes of elements of $G$). Now we have $$gh=hg(g^{-1}h^{-1}gh),$$ and $hg(g^{-1}h^{-1}gh)\equiv hg$ in the quotient group (since their difference is a commutator); so $gh$ - which is a representative of $ab$ - is equivalent in the quotient group to $hg$ - which is a representative of $ba$. So $ab=ba$.
You separately asked
Why is this so helpful if in fact we get weaker (poorer) group than the first one?
Well, this is a general fact about mathematical structures - just because one structure is "smaller than" another doesn't mean it's less interesting, and indeed the quotients and substructures of a given structure can be extremely useful in analyzing it. Moreover, specific quotient constructions like the abelianization show up in other contexts; a good example of this is the fact that the first homology group is the commutative version of the fundamental group (see this question for details).
And there are contexts where we have a natural algebraic structure, but we're only interested (for the moment) in certain aspects of it. Sometimes, for instance, for a particular context we may only be interested in the non-torsion part of a group. (Emphasis on sometimes, by the way; other times, the torsion part is really important!). Basically, "truncating" a mathematical structure lets you "focus" on the behavior that you care about. Sometimes this is a thing you want to do. Sometimes it's not.
