Concept of Linearity

For the sake of motivation, consider the function $f : \mathbb{R} \to \mathbb{R}$ given by $f(x) = ax$, well, it's pretty clear that $f(\lambda x) = \lambda f (x)$ and $f(x + y) = f(x) + f(y)$. What's the graph of this function ? A line. So this behavior is then called linear. Many things behave linearly and it's always related to behave like that.

You've said on comment that $f(x) = ax+b$ is not linear. Well, this is a translation of something linear, and it's called affine. It's a matter of terminology to choose calling the behavior of $f(x) = ax$ linear and not the behavior of $f(x) = ax+b$.

The main point is that this kind of behavior is found over and over again in math: functions, elements of $\mathbb{R}^n$, matrices, all of them combine with this kind of behavior with the usual operations. Linear algebra is then devoted to the systematic study of this property, generalizing the notion of a set on which elements can be combined linearly in the notion of a linear space. Calling those spaces vector spaces is just because the main motivation is the study of vectors (in the sense of geometric objects) on the plane and space. Althought that's the motivation which is used to start most of linear algebra courses, the reason we have linear algebra in mathematics is to have one unified and systematic way of studying this property: linearity. And believe, there are many consequences that come out from this single property.