On the philosophy of Category Theory I have been told by my professor that Category Theory is not just a language but is a shift in the way we think. As an example he pointed out that in category theory we do not worry about the objects themselves but he study the morphisms which relate it to other object. But what I do not understand is why is this important.
For example If I want to say a group is commutative then I take a map from $\beta :M \times M \rightarrow M \times M$ which changes $a.b$ to $b.a$. Then I take the usual binary operation map say $\mu $ and then the condition is equivalent to saying that $\mu =\mu \;\mathrm{o}\; \beta$. Can somebody explain why this is advantageous?
 A: $\beta$ is a map which makes sense in any symmetric monoidal category, and it can be used to define what it means to be a commutative monoid object in any such category. For example, if $(\text{Vect}, \otimes)$ is the symmetric monoidal category of $k$-vector spaces under the tensor product, then a commutative monoid object is precisely a commutative $k$-algebra. So one basic upshot of this kind of definition is that it can be internalized to different kinds of categories, which helps us see different definitions in mathematics as being different instantiations of the same more abstract definition.
That example is in some sense obvious, but here's a less obvious one. In homological algebra, algebraic topology, etc. you'll implicitly keep running into the symmetric monoidal category of $\mathbb{Z}$-graded vector spaces, by which I mean $\mathbb{Z}$-indexed sequences $V_n$ of vector spaces. The monoidal structure is given by
$$(V \otimes W)_n = \bigoplus_{i + j = n} V_i \otimes W_j$$
and so a monoid object is precisely a $\mathbb{Z}$-graded ring. Now what is the symmetric monoidal structure? An obvious symmetric monoidal structure given by
$$\beta : V_i \otimes W_j \ni v \otimes w \mapsto w \otimes v \in W_j \otimes V_i$$
with respect to which a commutative monoid object is precisely a $\mathbb{Z}$-graded commutative ring; these do show up, e.g. in algebraic geometry when defining projective varieties. But there is another more interesting symmetric monoidal structure given by
$$\beta : V_i \otimes W_j \ni v \otimes w \mapsto (-1)^{ij} w \otimes v \in W_j \otimes V_i$$
with respect to which a commutative monoid is a graded commutative or super commutative $\mathbb{Z}$-graded ring; this is called the "Koszul sign rule," and in many parts of mathematics it is totally inescapable. For example, cohomology rings are always graded commutative.
Now, you can learn about graded commutativity as a separate definition, but if you know that it's just commutativity with respect to a funny symmetric monoidal structure, then you can write proofs where you pretend that it's the same as ordinary commutativity, except that wherever you would ordinarily switch two elements you instead explicitly apply $\beta$. There are also lots of other kinds of structures you can internalize by using $\beta$ that automatically get the right signs put in to their axioms this way, e.g. Lie algebras. 
A: Writing out definition of a group, for example, in terms of arrows and commutative diagrams rather than sets and elements allows you to easily generalize the concept of a group in the category of sets to other categories. For example, a Lie group is just a group in the category of smooth manifolds, and a topological group is a group in category of topological spaces. 
See this nlab article for the complete definition of a "group object" in terms of arrows and diagrams.
