Ciro Santilli's preferred visualization of the real projective plane is a small variant of the standard "lines through origin in $R_{3}$".

Take a open half sphere e.g. a sphere but only the points with $z>0$.

Each point in the half sphere identifies a unique line through the origin.

Then, the only lines missing are the lines in the x-y plane itself.

For those sphere points in the circle on the x-y plane, you should think of them as magic poins that are identified with the corresponding antipodal point, also on the x-y, but on the other side of the origin. So basically you you can teleport from one of those to the other side, and you are still in the same point.

Ciro likes this model because then all the magic is confined just to the $z=0$ part of the model, and everything else looks exactly like the sphere.

It is useful to contrast this with the sphere itself. In the sphere, all points in the circle $z=0$ are the same point. But this is not the case for the projective plane. You cannot instantly go to any other point on the $z=0$ by just moving a little bit, you have to walk around that circle.