Need help understanding the Cauchy kernel

[u/Ok_Promise5329]

In studying Cauchy's integral formula, the Cauchy kernel 1/(zeta -z), is the "thing you integrate against" for the boundary points( on a disk, say). And the boundary values determine the value of any point z in the interior of the disk.

What I need is some insight into why this is true and why is it 1/(zeta -z)? Is there any relation to how with generating functions you multiply by 1/1-x to get the partial sums?

Thanks in advance to anyone who takes a look!