This can be proven algebraically,
$$\left|\frac{-a+e^{-j \theta}}{1-a e^{-j \theta}}\right|=\sqrt{\frac{(-a+\cos (-\theta))^2+(\sin (-\theta))^2}{(1-a \cos (-\theta))^2+(a \sin (-\theta))^2}}=\sqrt{\frac{a^2-2 a \cos (\theta)+1}{a^2-2 a \cos (\theta)+1}}=1$$
This is unsatisfying because the problem has a nice geometric meaning. Note that $\left|\frac{-a+e^{-j \theta}}{1-a e^{-j \theta}}\right| = 1 \iff \left|\frac{-a+e^{-j \theta}}{\frac 1 a - e^{-j \theta}}\right| = |a|$.
Geometrically, this is equivalent to proving that the blue length in this diagram is $a$ times larger than the green length.
I know that the $\frac 1 a $ and $a$ points have very useful properties in projective geometry, but I couldn't quite write a proof.