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The complex which computes Cech cohomology for a covering is the "same" one as the one that computes the cohomology of the nerve of the covering. It is not hard to see that the geometric realization of that nerve is, in some sense, an approximation to the original space. Since you apparently find it intuitive that simplicial cohomology detects the geometry, then this should convince you that Cech cohomology also does :P