In mathematics, it is common to define structures abstractly, i.e. purely in terms of certain algebraic properties that they satisfy, without specifying anything about what the objects themselves are. So there is no single notion of "vector addition". Vector addition can be any operation whatsoever, as long as it satisfies the rules of a vector space: associativity, commutativity, and so on. The operation you are likely familiar with, of tip-to-tail adding arrows in the plane (or in 3D space), is just one possibility. We can instead look at something like the vector space of polynomials, where vector addition is the same thing as ordinary addition of polynomials.

The strength of this abstract definition comes from the fact that it can apply to such different-seeming things as polynomials and arrows in the plane. Any fact we discover about vector spaces will apply to polynomials, to arrows in the plane, and indeed to any structure that satisfies the simple list of vector space properties. Because of this, the tip-to-tail addition of arrows can still be useful for intuition. While vectors can look like pretty much anything, they will act like arrows in the plane, at least when you're adding them or multiplying them by scalars.

As for why these specific rules: well, these are just the rules for the structure we call a vector space. There are many other mathematical structures out there. If you want to look at structures that are defined more strictly, so that you do have well-defined notions of "size" and "direction", you could look at inner product spaces. If you want to go the other way, dropping the operation of scalar multiplication so that structures are only required to have addition, you could look at abelian groups. Choosing a more specific definition pins down the structures more precisely, ensuring they will behave more similarly to each other, while choosing a less specific definition allows your definition to apply to more structures and makes your findings more general. It just so happens that the definition of a vector space provides a nice balance between these two extremes, which is why vector spaces turn up a lot in mathematics.