Belay jackets 101
The concept of a belay jacket has various origin stories, but American climber Mark Twight is typically credited 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 - when skiing, 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 a VERY warm jacket to put on quickly over everything when you stop.
British climbing expert and winter hardman Andy Kirkpatrick has a wonderful, deep dive article on his website that covers the history, theory, design features and recommended models for belay jackets. It’s a great read for all cold-weather athletes.
(Yes, it’s a few years old, but the general principles don't change much.)