I think it's kinda similar to some of our regular, everyday instinctual reactions. For example, our eyelids will instantly close when something might get into our eyes without us having time to think about it. If I was riding a bike and a bug accidentally almost gets in my eye, my eyelids will close before I can consciously think about closing them. If humans didn't have this reflex reaction then I would've been blind long ago, cause oftentimes there's no way you can consciously react fast enough to close your eyelid before something like glass or a bug gets in it. So I think his Spider-Sense is a very heightened part of his reflexes. Just like how breathing is a reaction that I don't have to consciously think about it, unless I decide to say, hold my breath.
Oftentimes, I think Spidey is purely reacting by instinct and not relying on conscious thought to avoid attacks or bullets. He's not the Flash where everything is slowed down for him and he isn't Batman or Captain America who are highly trained and know why they make the moves they make. This also doesn't mean he'll never get hit. Just cause he's knows where something (like a bullet) is coming from doesn't mean he can avoid it in time. Also, let's say MJ dies (although I'd rather not) and Peter decides he wants to kill himself by standing in front of a bullet then he could do that but similar to holding your breath he would have to consciously ignore his instinct to move out of the way (or to breathe). The most overused phrase I've heard Spidey use in regards to his Spider-Sense (other than the obvious one) is...."My body is screaming" for me to move out of the way or something to that effect.