Let $f(x,y) = u(x,y) e_x + v(x,y) e_y$ be a vector valued function on $\mathbb{R}^2$. I'm not sure how the formula is determined.
I take this definition of curl:
The curl at point $P$ is the (directed) anglular velocity a "infinitesemal" disc would rotate with, when it's center is aligned with $P$
From this, it's easy to see (by eg: treating it as torque) for a curve $\gamma$ with enclosed area $A$, the torque experined by it is $\frac{1}{|A|} \oint_{\gamma} f \cdot dS $ where the integral is taken along the surface of the curve.
Now, we need to prove that $\displaystyle \lim_{|A| \rightarrow 0} \frac{1}{|A|} \oint_{\gamma} f \cdot dS = \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}$ for any curve $\gamma$, but how do you do that ?
I'm stuck even for the very simple case when $\gamma$ is a circle, and I'm finding it very counterintuive why would the limit even be the same and not depending on the shape of $\gamma$.