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 $\mathbb{Z}$:

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

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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?

  [1]: http://etc.usf.edu/clipart/40500/40529/Pie_01-03a_40529_md.gif