Geometric Significance that 2D Points Form a Line

I'm reading through Multiple View Geometry in Computer Vision, by Hartley and Zisserman, and on page 2 it is stated that points at infinity in the two-dimensional projective space form a line, and in three dimensions they form a plane.

This isn't a mathematically rigorous question, but could someone help me to understand (at least geometrically) what this means?

• Some motivation comes from looking at any picture drawn with two point perspective. The vanishing points ("points at infinity") are all on the horizon: The horizon is the line consisting of points at infinity (and is thus "the line at infinity"). Stillwell's Four Pillars of Geometry has a nice section on perspective and projective geometry. – pjs36 May 25 '16 at 0:06

Part of the value of thinking about adding points "at infinity" to the Euclidean plane (so constructing the projective plane) and working with the projective plane is that in the projective plane there is symmetry between points and lines. Since each pair of points determines a line, each pair of lines should determine a point. For lines $L$ and $M$ that are parallel in the Euclidean plane we say that the (new) point $p$ at which they meet is "at infinity". Now suppose $q$ is a point in the Euclidean plane. There must be a line joining $p$ and $q$. I turns out to be the line through $q$ parallel to $L$. Now think about two points at infinity, on two nonparallel Euclidean lines. They must determine a projective line. The axioms all work out beautifully when that line contains precisely all the points at infinity.