The concept of a belay jacket has various origin stories, but many people credit American climber Mark Twight with popularizing the idea. Old school layering used to be something like: base layer, fleece or wool, hard shell on top. But with a belay jacket, you dress more like a minimalist cross-country skier - if you are out for a ski session, even when it's really cold, you don't need much in the way of layering, because you're maintaining your heat through movement. But if you stop for more than a few minutes, you better have something to put on.
Applied to climbing, you may start with a base layer and maybe a wind shell when moving. But in your pack is an extremely warm jacket to put on quickly over everything when you stop.
British climbing expert and winter hardman Andy Kirkpatrick has a wonderful article on his website that covers the history, theory, design features and recommended models for belay jackets. If you're thinking of getting one, or even have one already, this is a great read.