Since it can be shown that there is a bijection with lines through the origin and points on the projective plane, this means that if you trace families of parallel lines on the projective plane, that will correspond to the horizontal line through the origin or "point at infinity". It also shares the same direction with the parallel lines.

I imagine this when laying down thinking my arm is the 1-dim subspace and the ceiling is the projective plane, if I trace families of paralell lines, they all end up being the same line (I know this sounds stupid but it is an easy way for me to visualize it).

This also makes it clear where the general equation of lines come from. Imagine the same line through the origin [a:b:c] and another line presumedly perpendicular to it [x:y:z], then by Pythagoras it can be shown that ax + by + cz = 0, which is precisely the plane perpendicular to [a:b:c]. So if you imagine that plane also horizontal then that corresponds to the "horizon at infinity". The general equation of the line aX + bY + c = 0 is just the intersection with the projective plane with homogeneous coords (X,Y,1), for X = x/z and Y = y/z, z != 0. Points in the projective plane correspond to lines through the origin, and lines in the projective plane correspond to planes through the origin.

I always wonder why the projective plane is not the "default" plane that is taught in school when reality is seemingly projective.