In fixed aperture zooms, the thing that is fixed is the ratio of the focal length to the diameter of the aperture openning. To maintain the fixed ratio throughout the entire zoom range, the aperture is mecahnically linked to the zoom such that it changes its diameter with resect to the focal length setting.
The more complex mecahnism is the main reason for the higher price of "fixed" aperture zooms.
For modern SLRs where the lens is able to transmitt its zoom range and actual focal length setting to the camera body, the camera can actually maintain the fixed f-number for you instead of relying on the lens mechanism, so you can effectively have fixed f-number throughout the entire zoom range. However, the max aperture (min f-number) is limited.
For example, for a f/4 ~ 5.6 lens, if you set the camera to maintain fixed f-number when you zoom, you can get max f/5.6 throughout, but not f/4.