By "plane" they mean in the Euclidean geometric sense; the plane is infinitely wide, infinitely tall and, most importantly, infinitely thin.
The thickness of the painted line at the front of the goal line varies from field to field and even from point to point on the same field, especially on grass, so they go by the "plane of the goalline" which is positioned on the leading edge of the painted line instead.
The football crossing the leading edge of the painted goalline, even by just a smidge, is a touchdown.