\(\Pic(X) \cong H^1(X, \OO_X^*)\), and for \(\OO_X\)-modules \(\mcf, \mcg\) the extensions \[ 0 \to \mcf \to \mce \to \mcg \to 0 \] up to equivalence are classified by \(\Ext^1(\mcg,\mcf)\), which is \(H^1(X, \mcf \tensor \dualof{\mcg})\) when \(\mcg\) is locally free.
The isomorphism is the Čech description read backwards: a line bundle is trivial on some cover, the transition functions \(g_{ij} \in \OO^*(U_i \intersect U_j)\) satisfy the cocycle condition, and changing the trivialisations changes them by a coboundary. So \(H^1\) of the units is the group of line bundles, with tensor product as the group law.
This is the same phenomenon as the failure of right exactness on sections in stalks and exactness, and it is worth saying so: in both cases \(H^1\) measures the failure of local data to be global, once as sections that do not glue and once as trivialisations that do not agree.
On a complex manifold the exponential sequence makes it explicit, \[ H^1(X,\OO) \to H^1(X,\OO^*) \to H^2(X,\ZZ) , \] where the second map is the first Chern class: the topological type of a line bundle, with \(H^1(X,\OO)\) measuring how many holomorphic structures carry it.