What is the interpretation of homogeneous line intersection?

Consider three-dimensional space $\mathbb R^3$. If you normally homogenize by appending $z=1$, that means that your geometry as you know it happens on the $z=1$ plane in space. But it's also possible to view geometric elements as linear (i.e. containing the origin) subspaces of the whole three-dimensional space.

A point on the plane corresponds to a line through the origin. The line is described by any non-zero vector pointing along that line to give its direction. The point it represents is the point where the line intersects the $z=1$ plane. If the line is parallel to the $z=1$ plane, then the homogeneous coordinate vector has $z=0$ and both represent a point at infinity.

A line on the plane corresponds to a plane through the origin. The plane is described by any non-zero vector orthogonal to the plane, i.e. by a normal vector. The line it represents is the line where that plane intersects the $z=1$ plane. If both planes are parallel, it's the line at infinity with homogeneous coordinate vector $(0,0,1)$.

Incidence (point lies on line) corresponds to inclusion: A point lies on a line in the projective plane iff the line which models the point lies within the plane which models the line. Which in turn means that the two vectors describing these must be orthogonal to one another. That's why “dot product equals zero” can be used to check for incidence.

Now the key feature of the cross product is that the result is orthogonal to both the arguments. So you have two vectors $l_1$ and $l_2$, and both of them represent a plane in space through the origin and orthogonal to that vector. So the line of intersection is the set of all vectors orthogonal to both, and any one such vector can be used to describe the direction of the line.

@MvG has answered it and I accepted it. I just wanted to post a picture I've drawn for this to help me understand in case if someone else finds it useful.