This is a picture from my book. I extended the line M to get a better idea of where $p_n$ is.
It says the following:
It is natural to call the horizontal lines through O the "points at infinity".
What exactly does this mean?
If the points are on the z=-1 plane, how can line m extend to horizontal lines through the origin? I'm assuming horizontal lines through the origin means the xaxis, yaxis, or any lines in the plane z=0.
the horizontal lines do refer to the lines on z=0, they are called points at infinity as each line is represented by the point at which it intersects z=-1 plane but since lines laying on z=0 never intersect z=-1 they are said to intersect it at 'infinity'.