A lot of this has to do with the fact that you cannot feel velocity, you can only feel acceleration. So if you're walking and the coffee isn't spilling, you can increase your velocity at a constant rate (because you intuitively think it's the speed that matters). But when you stop, if you don't first slow down at a constant rate, then you're acceleration will change abruptly and you spill your coffee (because you intuitively think stopping won't matter).
