This is really simple when you think about it. The additional element changes the focal length of the lens, without changing the apparent size of the aperture. That means that the relative size of the aperture decreases, so the f number does in fact actually change. (If this is unclear to you, see the bit about f numbers in this other answer.)
This is also why real wide-angle converters can go the other way, effectively increasing the aperture. (See How can a speedbooster improve the light performance of a lens? for more.)
Some converters communicate intelligently with the camera body, so the aperture displayed will be correct. This is the case with the Canon extender you have, but might not be with third-party ones. This explains the part you were confused about: the camera is aware of the change already and the numbers it is showing you are what you will actually get. When you set the aperture on the camera to f/5.6, the aperture on the lens is set to the same position that would be f/2.8 without the extender (but which genuinely is f/5.6 with it).
Note that teleside converters which go on the front of the lens do change the effective aperture (see What's the difference between real and effective aperture?), so they don't change the f number. (They are usually lower quality, however, and can introduce vignetting.)