I know there are a lot of references (e.g. Griffiths-Harris page 141), but the issue is that these references always prove the proposition in arbitrary dimensions, using a somewhat contrived calculation involving Stoke's Theorem but I feel like there should be a much simpler explanation in the case of curves.
So let $X$ be a projective nonsingular curve over $\mathbb{C}$. We have the exponential sequence $$ 0 \rightarrow \mathbb{Z} \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_X^*\rightarrow 0$$ which gives us a long exact sequence in cohomology, and the first Chern Class is given by $H^1(\mathcal{O}_X^*) \rightarrow H^2(\mathbb{Z})\cong \mathbb{Z}$. Now a Cech 1-cocycle $\{g_{\alpha\beta}\}$ for a sufficiently fine cover of $X$ in $\mathcal{O}_X^*$ encodes the information of a line bundle, and based on what I have read the image of this in $H^2(\mathbb{Z})$ is given by $\{(\log g_{\alpha\beta} - \log g_{\alpha\gamma} + \log g_{\beta\gamma})/ 2\pi i \}$ where the discrepancies in the choice of logarithm give rise to the cohomology class. It seems to me that the poles and zero's of local sections used to define the $g_{\alpha\beta}$ should encode the "failure" to produce logarithms, but unfortunately I am lost as to where I should go from here. Any help, including an explanation about why this is doomed to fail will be highly appreciated!