$H^1$ is the first derived functor of the functor $H^0$ of global sections.
In the Cech cohomology construction, note that we look whether the local sections glue together to form global sections. On an affine space, this is indeed true. But, not so in general. We would like to address the obstruction using algebraic means. When things glue well, we have an exact sequence. In homological algebra, the question of exact sequences breaking down under a functor is addressed by the machinery of "derived functor".
So $H^1$ and the higher cohomology groups are in a sense the obstruction to local sections patching up to form global sections. Since $H^0$ behaves well on affine spaces, it in a sense measures the failure of affineness also. It captures geometry by seeing how the affine pieces glue together to form a projective variety, for instance. The dimensions in which geometry is interesting can be seen by at which dimension the derived functors are nontrivial.
This is my personal point of view to see how geometry is involved, based on derived functors.