In the page you reference, there is a sentence about this: " It is not possible to use transparency (alpha) with background colors on the main drawing surface; you can acheive the same effect with createGraphics()."

Even though "officially" Processing's main canvas layer is 100% opaque,
we can force it to have any "alphaness" we wish!

Here's a function snippet on how to do it:

void prepareCanvas() {
  // Loads canvas' PGraphics g into pixels[]:
  loadPixels();

  // Makes PGraphics g 100% transparent black:
  Arrays.fill(pixels, 0);
  updatePixels();
}

And to see an example using it, just go to the post link below:
http://forum.processing.org/topic/wanna-change-direction-with-keypressed

" we can force it to have any "alphaness" we wish!"
If you don't make the window of the sketch itself transparent, I fail to see the point.

Mmm, looking closer at the sketch, I see it is a hack, since after drawing on this transparent background, you transfer the result to a PGraphics. The result is the same than drawing the bird directly on a separate PGraphics.

There is another trick for that:

1. PImage imgBG;
2. PGraphics imgBird = createGraphics(w, h, JAVA2D);
3.  
4. // [...]
5.  
6.   // Draws bird in the canvas:
7. //  prepareCanvas();
8.   PGraphics gg = g;
9.   g = imgBird;
10.   imgBird.beginDraw();
11.   drawBird(280, 272, 200, 20, 20);
12.   imgBird.endDraw();
13.   g = gg;
14.  
15.   // Copies PGraphics g to PImage imgBird:
16. //  imgBird.copy(g, 0, 0, w, h, 0, 0, w, h);

A Java2D PGraphics is already transparent. No need to fill the whole sketch surface for that.

Imagine that each frame of your animation is a page in a flick-page animation (you know, the sort where you draw a picture in the corner of each page of a pad of paper and flick through them to make them more).

Perhaps what you haven't understood is that most functions of Processing, including background(), can be called as it, or on a PGraphics.
background() with alpha on the main sketch surface is mostly useless, because there is still the background of the sketch window behind it.
But you can call someGraphics.background() too, and there the alpha might have sense.
I think that's the point of the sentence I quoted.