A brake was originally a horse-drawn vehicle. I do not know the origin of the word. A shooting brake was used to transport shooting parties around an estate. Cars began to be developed for the same purpose and were also called shooting brakes. I thought the term had long been replaced by estate car.
My OED thinks it might come from an old word for a cage, 'brake' (not related to the brakes that stop a car), but isn't sure. If the OED isn't sure, I suppose nobody is.