The intuition is that of a landscape painter: the projective plane adds the points on the horizon to the usual plane when viewed, like here
http://www.amazon.com/Landscape-Painting-History-Nils-Buttner/dp/0789209020
from slightly above. From the point of view of an observer floating slightly above the plane, lines of sight come in two flavors: those that intersect the plane below her, and those that do not, corresponding to the points on the horizon ("at infinity" in the usual mathematical jargon). Each such line of sight corresponds to a point in the projective plane, and this connects with the usual mathematical description of the projective plane as the set of lines through the origin in 3-dimensional space.