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.
When you build three dimensional projective space from three dimensional Euclidean space the same kind of argument leads to the plane at infinity.
I suspect that in thinking about computer vision you often have to manipulate a projection of some solid figure onto a plane. If you work in projective space you never have to treat parallel lines as a special case. Look ahead past page 2 in your book for places where something like this comes up.