I am going to start by saying I do not find Green’s theorem very intuitive. It is a magnificently powerful theorem. It has been proven, so it must be true, but unlike many other theorems, I do not feel it comes naturally. (I have mentioned this before!) But I think that is because it does not start with our normal x-y Cartesian plane.
There is a really good discussion of this on the Khan Academy website, which is the best explanation I have seen. The gist of it is that the relationship of a closed curve to the area it encloses is analogous to the relationship of a curve and the area beneath that curve from freshman calculus. Having such a relationship is powerful because we can use it to find the area enclosed in any two-dimensional shape using a calculation on the length of the line around it.
That is where the planimeter comes in. The planimeter is a device to measure the line circumscribing a shape. It also performs the calculation. The process is beautiful to watch and you can see an example here: