Notice that $g(t), 0\le t\le 2\pi$, is a closed curve in the plane. It is parametrized with respect to arc length, and its curvature is given by $|f'(t)|$. By assumption, $a\le |f'|\le b$.
For convenience, let's discuss the case $x=0$. I claim that the inequality may be viewed as the condition that we will still be able to return $g(0)$, given the restrictions on the curvature.