Let $\alpha(s)$ be a unit speed, simple, closed curve. Then let $\gamma(s)=\alpha^\prime(s)$. Thus, $\gamma$ is a closed curve on the unit sphere. So, we would like to apply Crofton's formula to $\gamma$.
Crofton's formula says that $4L = {\int \! \!\int}_{S^2} n(W)dS$
where $n(W)$ counts the number of times that the curve $\gamma$ intersects the equator with $W$ taken to be the north pole, and $L$ is the length of the curve. So, $L = \int \| \gamma^\prime \|ds$
But, $\| \gamma^\prime \|$ is just $\kappa$, the curvature of $\alpha$. Thus, we end up with $\int \kappa ds = \frac{1}{4} {\int \! \!\int}_{S^2} n(W)dS$
Suppose we knew that $n(W)\ge 2$ for all points $W$ on the sphere for this curve. Then we would get, $\int \kappa ds \ge \frac{2}{4} {\int \! \!\int}_{S^2} dS$ $\int \kappa ds \ge \frac{1}{2}(4 \pi)$ $\int \kappa ds \ge 2\pi$
Which is exactly what we want to show! Can you prove what I claim about $n(W)$?