On $\mathbb{CP}^1\ni(X_1:X_2)$ let $z=X_1/X_2$ be one of the two charts, and define an involution map $$I^-:z \mapsto -\frac{1}{\overline{z}}.$$
Question: how to prove that the quotient $\mathbb{CP}^1/I^- \cong \mathbb{RP}^2$ ?
Another related issue: is there a simple way to prove that, up to diffeomorphism, the only involutions of $\mathbb{CP}^1$ are $z \mapsto \pm \frac{1}{\overline{z}}$ ?
I'm just answering your main question, from the perspective of projective geometry. The map $z\mapsto\frac1{\bar z}$ is the inversion in the unit circle. But you are not only inverting, but then also reflecting in the origin. So points on the unit circle will be paired with points on the unit circle in the opposite direction. Points inside the unit circle will be paired with points outside. The origin will be paired with the point at infinity.
Now you can perform a stereographic projection of the complex plane onto the Riemann sphere. You will find that your paired points end up as antipodal points on the sphere. And the set of all points on the sphere, with antipodal points identified, is just the real projective plane. You can simply take the 3D coordinates of these points as homogenous coordinates, and the antipodal point will be the representant which has all coordinates negated and thus represents the same point, so the pairing fits in with homogenous coordinates there.