First, an important remark. Vectors are mathematical objects, just like numbers, matrices, tensors, groups, manifolds, etc. These things are a part of physics because they happen to be a good model for the physical world, but they are not a part of it. Vectors are not something that exists in physical reality, whatever that might be. So let's separate the math and the physics.
You can define a vector as an ordered tuple of real numbers, or as an arrow in space; it doesn't make any difference because the definitions are equivalent. In the case of an ordered tuple of numbers, addition is defined component wise. If you define a vector as an arrow, then addition is defined either with the parallelogram or the triangle law. These notions of addition can be shown to be equivalent, but I don't think that's the point of your question.
On the other hand, we find that a good way to model the way things move is with forces. Suppose you're floating in empty space near a star. You will feel the force of gravity pulling you towards the star. We can model this force as a vector, where the magnitude and direction of the vector correspond to those of the force, and you can use your knowledge of this force (so far there's only one) to calculate your movement using Newton's law.
Now suppose the star magically disappears and is replaced by a different one, maybe with a different mass and a little bit to the side. Now the force (and its associated vector) points to the new star, and you can again use Newton's law. But what if both stars are present? So far all we know is that it's convenient to represent a force by a vector, but we have no idea what to do when there are various forces. It is found experimentally that when there are multiple forces acting on you, it's exactly as if there were a single force, whose vector is given by the vector sum of the individual forces' vectors.
Although this sounds obvious, it is a nontrivial statement. A priori, who says that multiple forces acting at once are equivalent to a single force? It happens to be true, and that is the reason we say that forces are vectors.
Of course, I've used forces as an example, but the superposition principle also applies to velocities, positions, electromagnetic fields, etc. Because it is found experimentally that the superposition principle is valid, we use vectors to model all those things.