Green's theorem essentially gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. What I don't understand is, while a line integral along a curve C geometrically gives the area of the 'wall' between the curve and a function, a double integral over a region D gives the volume between a surface in space, and the region D. How are they compatible?
First think of a simpler situation, i.e., the fundamental theorem of calculus. One form of it is that $$\int_a^b f(x)dx=F(b)-F(a),$$ where $F$ is any anti-derivative of $f$. Now the left hand side can be interpreted geometrically as the area under the graph of $f(x)$ over the interval $[a,b]$ while the right hand side is just the evaluation of a function at two points. So in this situation I guess you could ask again "How are these two things compatible?" as you did with Green's theorem. But this is precisely what makes this formula particularly nice (and what makes a "nice" formula in general), i.e., it shows that two seemingly unrelated things are in fact related, which is what mathematics is all about. I think Hilbert once said something to the effect of "Poetry is the art of giving different names to the same thing, while mathematics is the art of giving the same name to different things".