Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault cohomology, just by putting $\mathcal{F}=\mathbb{R}$ or the sheaf of holomorphic functions $\Omega^p(M)$. But each generalisation involves losing intuition about the measured geometric properties.

My question is: **is it possible to understand the geometric intuition behind sheaf and Cech cohomology in the same way one can understand the geometry behind de Rham cohomology?**

See also this related question.