Expanding on the comment of RBega2:

Let $(x(t),y(t))$, $t\in [0,1]$ be a parametrization of $C$. From Green's Theorem,
$$\int_C-y\,dx+x\,dy=\iint_R2\,dxdy=2\cdot Area(R).$$
From Cauchy-Schwartz inequality,
$$\vert\langle x,y\rangle\cdot\langle -\dot y,\dot x\rangle\vert
\leq \Vert\langle x,y\rangle\Vert\,\,\Vert\langle -\dot y,\dot x\rangle\Vert
=(x^2+y^2)^{1/2}(\dot x^2+\dot y^2)^{1/2}.$$
Therefore, we have
\begin{align*}
2\, Area(R)=
\int_Cx\, dy-y\, dx&=\int_0^1x(t)\dot y(t)-y(t)\dot x(t)\, dt \\
&\leq
\int_0^1(x^2+y^2)^{1/2}(\dot x^2+\dot y^2)^{1/2}\, dt \\
&=\int_Cr\, ds.
\end{align*}
Equality holds if and only if $\langle x,y\rangle$ is parallel to $\langle\dot y,-\dot x\rangle$, that is if $\langle x,y\rangle$ is orthogonal to the velocity vector $\langle\dot x,\dot y\rangle$ which is exactly when $C$ is a circle centered at the origin.