You can always find a way to translate the abstract viewpoint back into down-to-earth phrasings, but the point of abstraction is that this is *not the right thing to do* because you might do things and think "Hm, all of those look kinda similar". In the abstract approach, you do the proof *once* and you are done. You think "Hm, there's something about zeros and poles for Cousion problems, and something about zeros and poles for the (algebraic) Riemann-Roch theorem, I wonder what's the connection".

Let me try to motivate line bundles generally: If you view a manifold as a collection of states for a system, then the notion of fibering a manifold corresponds to attaching ideas such as colour, speed, temperature, humidity to the state of the system.... Now, a bundle is a special kind of fibering, a locally-trivial fibering.
(*How do I understand this in terms of, say, colour*?)
Anyway, I'm guessing a line bundle is just a way to say we're just attaching velocity to the point, similar to how a line bundle to a function like $y = f(x)$ is just the collection of points plus tangent lines?