Think of the projective plane as half a sphere, border included, with antipodal points in the border circle identified. A projective line is then half a great circle with the points in the border circle identified.
If you want to think of the "plane" model, think you are in R^3, project, say, the unit sphere into the x,y-plane.Then you can think of the projective plane as being obtained, via orthogonal projection, from a unit disk with antipodal border points identified. Projective lines are then the projections of the half circles mentioned above with the border points identified.
Hope my english makes it clear.