From the definition of Möbius transform we get that the image $w_k$ of a point $z_k$ is $$ w_k = \frac{az_k+b}{cz_k+d} $$ From this we get $$ w_i - w_j = \frac{az_i+b}{cz_i+d} - \frac{az_j+b}{cz_j+d} = \frac{(ad-bc)(z_i-z_j)}{(cz_i+d)(cz_j+d)} $$ and by inserting this into the cross-ratios $$ \frac{(w_1-w_2)(w_3-w_4)}{(w_1-w_4)(w_3-w_2)} = \frac{\frac{(ad-bc)(z_1-z_2)}{(cz_1+d)(cz_2+d)}\frac{(ad-bc)(z_3-z_4)}{(cz_3+d)(cz_4+d)}}{\frac{(ad-bc)(z_1-z_4)}{(cz_1+d)(cz_4+d)}\frac{(ad-bc)(z_3-z_2)}{(cz_3+d)(cz_2+d)}} = \frac{(z_1-z_2)(z_3-z_4)}{(z_1-z_4)(z_3-z_2)} $$