As I understand it, the ISO value indicates how sensitive the sensor/film is to light. How is this related to speed? Is it something to do with how fast old chemical film developed?
In photography, how quickly something happens is usually referred to as speed, irrespective of the mathematic time/distance definition. That shorthand is also why we have 'fast' lenses because they enable an exposure to be made more quickly.
Hence the 'speed' of a film/sensor is how quickly an exposure could be formed, rather than development time. Films were compared to one another as being 'faster' and 'slower' with the physical properties of each offering a tradeoff of detail against the ability to freeze motion and in digital it is usaully a tradeoff of accuracy against noise.
ISO 12232:2006 Photography -- Digital still cameras -- Determination of exposure index, ISO speed ratings, standard output sensitivity, and recommended exposure index defines the 'ISO' for digital sensors but there never was just one standard. There are a range of separate standards for combinations of process (negative/positive), media (film/paper), colour/mono as well as by type:- there is one for black and white aerial camera films for example.
The concept of light sensitivity is related to speed in that a faster film (higher ISO value) requires less time exposed to light than a slower film (lower ISO value) for a single exposure. Fast film achieves this by having larger crystals of silver salts than slower film, thus collecting more light and creating larger "grain". "ISO speed" might better be called "equivalent ISO speed", given that the ISO numbers refer to wild things like ISO 12800 for which no film exists.
The value relates to the shutter speed to achieve a satisfactory exposure in a particular ambient light situation. "Fast" film will allow a fast shutter speed, therefore being more suitable to capturing movement, or indirectly, low light situations. You trade off the graininess of the image, for capturing the image at all. Think of graininess as somewhat like pixels in digital cameras. A high ISO number has less pixels in the image.
Or another way - a photo lets a specific bucket of photons through the shutter depending on how bright the light is, and how long the shutter is open. These photons have to service a fixed number of receptive crystals. Faster film has less crystals, slower film, more crystals. So slower film needs a longer shutter open time to get enough photons through the lens to service the larger number of crystals. But those crystals are finer and smaller and result in a better quality image. Digital cameras mimic the behaviour of film by effectively allocating photons to the available pixels.