Why does a camera's maximum (allowed) aperture get smaller when you increase its zoom?
The short answer is because it is cheaper to manufacture such lenses. The longer the lens and the wider the aperture, the larger the optical elements in the lens - thus larger the expense to produce them.
A lens like 70-200/2.8 must have a front optical element of 200mm/2.8=72mm, which is quite a chunk of glass. On the other hand, the 70-300/4-5.6 needs to be 300mm/5.6=54mm wide. If it were f/4 through its full range, the optical element would need to be 75mm wide - even larger than the much more expensive 70-200/2.8.
In your question, you say "the camera's maximum aperture". The camera does not have an aperture - the lens does. Minor but important difference, especially for SLRs - once you remove the lens you see that the camera is just a light bucket with a big hole in the front.
The aperture is the ratio of the focal length of the lens to the size of the front optical element. Essentially
aperture = focal length / optical element size
For example, a 50mm f/1.8 lens has a 28 mm (50/1.8) element size.
If you're wondering why the f-stop numbers don't seem to be linear (they're not), it is because the amount of light collected by the lens is proportional to the focal length divided by the aperture squared. Because of this power of 2, f/4 collects twice as much light as f/5.6, since 5.6/4=sqrt(2).
F-number is the ratio of aperture opening and the focal length. Imagine an aperture which is fixed in its diameter and located just behind the front element of the zoom lens. regardless of the actual focal length of choice. In this case, as you zoom in (increase FL), the ratio is getting smaller (the F-number is getting bigger). As you zoom-out, the ratio increases (the F-number is getting smaller).
In fixed-aperture lenses, the optical/mechanical design is such that the ratio is maintained throughout the zoom range. This allows zooming in and out while composing w/o worrying the you stopped down the aperture while out, when you need max aperture when you're back in.