I read that question on MSE. And there is such place
On each such set it is easy to see that $O_X(D)(U)\longrightarrow O_X(U)\longrightarrow L(U)$ defined by $t\longmapsto t\longmapsto ts$ is an isomorphism, so in the end we find that $O_X(div (s))\approx L$ .
for me it's not very clear why $t \mapsto t \mapsto ts$ must be isomorphism or morphism at all. And, even if it is isomorphism, why we need to conclude that $O_X(div(s)) \approx L$? Set of local isomorphisms it's not yet isomorphism between sheaves.
Hope for your help!