I'm somewhat new to algebraic geometry. I'm currently studying algebraic curves primarily over closed fields, (for future discussion let's just call such a curve C). I was taught to think of things like $\rm{Pic}^{0}(C)$ as a set of equivalence classes of divisors on C. From what I gather, these are usually called Weil Divisors.
Some literature seems to take the approach that these classes on $\rm{Pic}^{0}(C)$ can be thought of instead as a line bundle. For example, if I let $\epsilon$ be a two-torsion point on C, I can think of $[\epsilon]$ as an order two line bundle.
My question is what the connection is. How can I intuitively see that this must be true?