= Solution
Let $\mathcal C^\infty_{\mathbb C}$ be the additive sheaf of complex-valued <smooth functions>, and $\mathcal C^{\infty,*}_{\mathbb C}$ its multiplicative sheaf of nowhere-zero functions. The <smooth exponential sequence> is
$$
0\longrightarrow\underline{\mathbb Z}\longrightarrow\mathcal C^\infty_{\mathbb C}
\xrightarrow{f\mapsto e^{2\pi if}}\mathcal C^{\infty,*}_{\mathbb C}\longrightarrow1.
$$
Its kernel is the integer-valued locally constant functions, namely the integer <constant sheaf>. It is surjective on stalks because a nowhere-zero smooth function has a smooth logarithm on a sufficiently small neighbourhood.
The additive smooth-function sheaf is a <fine sheaf>, using a <partition of unity>, and the <smooth manifold> is a <paracompact space>. Its positive-degree <sheaf cohomology> vanishes. The <long exact sequence in sheaf cohomology> consequently gives an isomorphism
$$
\delta:H^1(X,\mathcal C^{\infty,*}_{\mathbb C})\xrightarrow{\sim}H^2(X,\mathbb Z).
$$
To connect this with bundles, trivialize a smooth <complex line bundle> over an open cover. Its transition functions $g_{ij}$ form a multiplicative <Čech cocycle>, satisfying $g_{ij}g_{jk}=g_{ik}$. Changing trivializations changes it by a <Čech coboundary>. Conversely, any such cocycle glues trivial line bundles, and cohomologous cocycles give isomorphic bundles. Thus the isomorphism classes are $H^1(X,\mathcal C^{\infty,*}_{\mathbb C})$.
The connecting isomorphism is the <First Chern class>. Locally choose logarithms $g_{ij}=e^{2\pi if_{ij}}$; on triple intersections
$$
n_{ijk}=f_{ij}+f_{jk}-f_{ik}\in\mathbb Z
$$
represents that class. Combining the gluing classification with $\delta$ gives the requested bijection:
$$
\boxed{\{\text{smooth complex line bundles on }X\}/\cong\ \xrightarrow{\ c_1\ }\ H^2(X,\mathbb Z).}
$$
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