We know that there is only one linear fractional transformation that maps three given points $\displaystyle z_1, z_2, z_3 $ to three specified points $\displaystyle w_1, w_2, w_3 $. So the mapping $\displaystyle \text{Im} \ z = 0 $ onto the circle $\displaystyle |w| = 1 $ is uniquely determined if we choose the points (for example): $\displaystyle z = 0, \ z = 1, \ z = \infty $.

Why do we write $\displaystyle w = e^{i \alpha} \frac{z-z_{0}}{z-z_{1}} $. Where did the $\displaystyle e^{i \alpha} $ come from?