I understand well the notion of a line bundle $L \to X$ where $X$ is the base. Locally at some $U \subset X$ we have that $L = U \times \mathbb{C}$ (if we work over the complex numbers).
I also understand the notion of the structure sheaf $\mathcal{O}_X$ and Serre's twisting sheaf $\mathcal{O}_X(d)$ in terms of regular functions over $U$ and in terms of rational functions of degree $d$ over $U$ respectively.
What I struggle to understand is how to understand the connection between sections of the line bundle with those sheaves though.