If A is an element of surface area in bounded by a simple closed curve C, let P be an interior point in C and $\mathbf{n}$ a unit normal at P.
By Stokes' theorem, $\iint(\nabla\times \mathbf{F})\cdot \mathbf{n}~dS = \oint_{C} \mathbf{F} \cdot d\mathbf{r}.$
Using the mean value theorem for integrals* we can write this as
mean[$(\nabla \times \mathbf{F} )\cdot \mathbf{n}$] =$ \frac{\oint_{C} \mathbf{F} \cdot d\mathbf{r}}{\Delta A},$ and the result follows from taking the limit as $\Delta A \to 0. $
In words, the expression for (curl$\cdot\mathbf{n}$) reaches a limiting value as the area A shrinks around the point P.
*The MVT for integrals is:
If a function f is continuous on [a,b] there exists a point c on [a,b] such that $f(c) = \frac{1}{b-a}\int_a^b f(x)dx.$