Trying not to get too technical, the f/stop is, as you say, a ratio of the focal length. This ratio is the relative aperture of the lens and describes the physical aperture of the lens. For example, a 50mm lens at a relative aperture of f/2.8 has a phyisical aperture of 50/2.8 = 17.86mm, and a 200mm lens at a relative aperture of f/2.8 has a physical aperture of 200/2.8 = 71.43mm. For the same f/2.8 relative aperture, the 200mm lens has a physical aperture four times that of the 50mm lens (speaking in terms of aperture diameters).
Okay, if an f/2.8 200mm lens has an aperture diameter four times that of an f/2.8 50mm lens, why do they both admit the same amount of light?
This is because a lens' aperture is an area phenomenon, and the area captured by a lens—its angle of view— is determined by a lens' focal length and the size of the sensor/film.
Assuming the same sensor, when pointed at a blank, evenly lit sky, a 50mm lens sees 16× as much sky as a 200mm lens. The sky radiates a given amount of light per square meter. The 50mm lens therefore gathers 16× as much light as the 200mm lens. This means that to pass the same amount of light through the lens, so as to have the same effect on the film or sensor, the 200mm lens must have an aperture that is 16× the area (4 times the diameter) than the aperture of the 50mm lens. This is the relative aperture, measured in f/stops, and does exactly that (as shown above).