In the book of S. Kobayashi, hyperbolic complex spaces, there is the lemma (3.A.3) on the page 153, section Royden's extension lemma, whose statement is
Every exact sequence of holomorphic vector bundles $0\to E'\to E\to E''\to 0$ on a Stein manifold splits.
The author's argument is that the obstruction to a splitting lies in $H^1(X,\mathrm{Hom}(E'',E'))$ which is zero since $X$ is a Stein manifold.
Could someone explain what obstruction means and why it lies in this cohomology group? Thank you very much.