Let $R$ be a ring, associative and unital, and let $A$ be a left $R$-module. Let $x\in A$ and define the complex $K(x)$ to be the complex $0\to R\to R\to 0$ where the middle map is multiplication by $x.$ The homology of $K(x)\otimes A$ are the Koszul homology modules $H_q(x,A)$ and the cohomology of the complex $\mathrm{Hom}(K(x),A)$ are the Koszul cohomology modules $H^q(x,A).$ For $\mathbf{x}=(x_1,\ldots, x_n)$ we define $K(\mathbf{x})$ to be the total complex of the tensor product $K(x_1)\otimes\cdots\otimes K(x_n)$ and then define $H_q(\mathbf{x},A)$ and $H^q(\mathbf{x},A)$ similarly.
We also define the local cohomology (of $A$ at $\mathbf x$? I am not sure of the terminology here) to be the direct limit $$H^q_\mathbf x(A)=\varinjlim H^q(\mathbf x^i, A).$$
My question is this: what, if any, is the geometric meaning behind these modules? The general theory is all well and good, but I feel that when I'm done studying these things they will just slip out of my mind because I have nothing to connect them to. It looks sort of like localization to me, kind of like how the localization of a ring $R$ at an element $x\in R$ is the direct limit of the system $R\to R\to R\to\cdots $ where each map is multiplication by $x.$ I don't have much experience in algebraic geometry however (especially from the scheme-theoretic point of view) so I can't put this into words myself.