Texgen modes GL_OBJECT_PLANE and GL_EYE_PLANE are generating texture coordinates based on the distances of vertices from a plane.
A plane in 3 dimensions can be defined by the equation:
Ax + By + Cz + D = 0
This is an implicit formula - it defines a plane as the set of points that satisfy the equation. In other words, it serves as a test to determine whether or not a particular point [x y z] is on the plane or not.
Another interpretation is that the value of
Ax + By + Cz + D
is proportional to the distance of the point [x y z] from the plane.