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There exists an exact sequence $0\rightarrow \mathbb{Z}\rightarrow\mathbb{C}\rightarrow\mathbb{C}^*\rightarrow 1$ defined by the expontial map which induces an isomorphism $H^2(M,\mathbb{Z})\rightarrow H^1(M,\mathbb{C}^*)$, this gives the correspondence between $H^2(M,\mathbb{Z})$ and line bundles which are classify by $H^1(M,\mathbb{C}^*)$ via the identification with Cech cohomology.