I am able to show that if a curve lies in a plane then it's curvature at a point $p$ is $$\kappa=\lim_{\mu\to 0}\frac{\sigma}{\mu}$$ where $\mu$ is the length of a segment of the curve containing p, and $\sigma$ is the length of the arc on $S^1$ formed by translating the velocity vectors of this segment to the origin.
Analogously, it can be shown that the Gaussian curvature at a point p on a surface, is given by $$\lim_{A\to 0}\frac{A^\prime}{A}$$ where $A$ is the area of a region containing $p$ and $A^\prime$ the area of the corresponding region on $S^2$ under the Gauss map. (Apparently this is how Gauss defined the curvature)
We also know that the Gaussian curvature is the product of the principal curvatures.
But the principal curvatures are the curvatures of plane curves by definition (curvatures of normal sections). Hence the principal curvatures are given by the first limit above.
So I guess this means that $$\lim_{A\to 0}\frac{A^\prime}{A}=\left(\lim_{\mu_1\to 0}\frac{\sigma_1}{\mu_1}\right)\left(\lim_{\mu_2\to 0}\frac{\sigma_2}{\mu_2}\right)$$ where $\mu_1,\mu_2,\sigma_1,\sigma_2$ correspond to the two normal sections.
How does this happen? Does this mean that the area $A^\prime$ is equal to the product of the two arc lengths $\sigma_1\sigma_2$? It's definitely not true if the region on the sphere is a cap!
The formula is correct, but of course you have to interpret it correctly, as a limit. The principal directions are at right angles, so you're looking at infinitesimal rectangles, both in the domain and in the range. (Also, remember that Gaussian curvature can be negative, and so we have to take orientation of the rectangle in the image into account.)
We're not saying that the area of a spherical cap, is per se, any such product. We're talking about an infinitesimal piece, say coming from a bit of arclength $\mu_1=\Delta\phi$ on the longitude circle and $\mu_2=\sin\phi\Delta\theta$ on a latitude circle. These get multiplied, respectively, by the principal curvatures, so that $\sigma_1 = k_1\mu_1$ and $\sigma_2=k_2\mu_2$. No contradictions that I can see, but ask further.