"Plane" simply means a geometric plane - A flat surface, or 2d slice of 3d space http://en.wikipedia.org/wiki/Plane_(geometry) . Other uses may be adaptations - but they all mean with basic geometric idea.
I have never heard "plane" used to mean "facade", but it could mean that if you were talking about the face of a building as a flat plane structure.
In your sentence, "The light glanced mirthfully through the dense leafage of the superb planes which overhang the dark-green water" "planes" is used to describe the trees. Branches and leaves often group in space and fan out, forming structures that are kind of flat like planes. If they are layered in space, each layer could also be thought of as a plane. This image of a bonsai is a great example of planes that occur in plants http://upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Bald_Cypress,_1987-2007.jpg/200px-Bald_Cypress,_1987-2007.jpg .
Planes are even used in art to simplify all sorts of forms when drawing and painting. http://www.jaimetreadwell.com/Draw-self-portrait-14.jpg With this process complicated and round shapes can be thought of as a bunch of small simple planes that when put together create much more form.
And then there is the word "plain" which has it's own variety of meanings but sounds exactly the same as "plane"....