I think it's pretty intuitive how singular/simplicial cohomology detects "holes" in a space.

How can we directly visualize how and in what sense the Cech cohomology of a cover does this?

In case it's of any interest, here are two examples I've looked at with the constant sheaf $\mathbb{Z}$:

(1) The disk, covered "Venn diagram style" with three open patches $U_1, U_2, U_3$ overlapping near the center (like this, but with overlaps), and

(2) The restriction of this cover to the boundary circle of the disk: three opens $U_1, U_2, U_3$ with 3 double intersections $U_{12}, U_{13}, U_{23}$ and *no triple intersection*.

If you look at the Cech complex in (2), the $H^1=\mathbb{Z}$ "comes from" the fact that you can write down a triple of elements $(1,0,0)$ on $U_{12}, U_{13}$,$U_{23}$ which "would" disagree on the triple overlap in (1), but since it's "missing", $(1,0,0)$ gets counted as a cocycle, which is not a coboundary. Even better, the presentation of this $H^1$ you get from the Cech complex is $\mathbb{Z}^3/\{(a,b,c)=(b,c,a)\}$, which is isomorphic to $\mathbb{Z}$ because you can "rotate" all the coordinates "around the missing intersection" into the first component.

I think a like minded analysis of higher dimensional analogues provides similar intuition. Are there any formulations of the Cech complex to really make precise how this intuition should work? What's going on here?

**Follow up:** Following Mariano's answer below, I started reading about Abstract simplicial complexes and their cohomology, which seem like just what I was looking for. What helped me most were the ideas that

1) The (constant sheaf) Cech cohomology of a cover $\cal{U}$ of $X$ "is" the simplicial cohomology of its nerve

$N(\cal{U})$, an abstract simplicial complex,

2) The simplicial cohomology of an abstract simplicial complex "is" the singular cohomology of its geometric realization, and

3) The geometric realization of the nerve of a covering of $X$ is a "simple approximation" of $X$,

So in this sense, we can say precisely that

Cech (constant sheaf) cohomology on a cover detects holes in a "simple approximation" to $X$ defined by that cover.

In particular, seeing the faces of a simplicial complex encoded as formal wedge products of its vertices totally made my day :)