Why composition is so important in category theory? I'm reading "Category: The Essence of Composition" 
As a software developer, I understand why composition is important in programming. It's allows you to get complex components from simple components, helps to improve readability and maintainability of software.
But why composition is important in math? I did not find an answer to this question neither in the above article, nor in Wikipedia.
In other words - that would be impossible if the composition "disappear"?
I'm not a mathematician, so I would appreciate for the most simple answer.
 A: Historically, category theory was created to encode the functoriality of homology. Given a natural number $n$, to any topological space $X$, you can associate an abelian group denoted $\mathrm H_n(X)$ which gives you some kind of information on the space $X$. As such, it does not seems like an overwhelming construction. What matters in that construction is that to any continuous map $f \colon X \to Y$ you can associate a morphism of groups $\mathrm H_n(f) \colon \mathrm H_n(X) \to \mathrm H_n(Y)$ in such a way that
$$ \mathrm H_n(h\circ g) = \mathrm H_n(h) \circ \mathrm H_n(g)
\qquad \text{(whenever the composition makes sense)}.  $$
In a modern language, one would say that $\mathrm H_n$ is a functor $\mathsf{Top} \to \mathsf{Ab}$.
So, for the first categoricians (Eilenberg and MacLane if I remember correctly), category theory is all about composition.

Another answer to you question could be the following: category theory without composition already has a name and is well studied. It is called (reflexive directed) graph theory.
There actually is a way to categorically say that category theory is graph theory with composition, but you emphasize that you want a simple answer and it would lead us a little to far.
