I understand that getting bigger is about the right combination of training/diet/rest. But when it comes to training, what really makes you grow? My question is does one need to train above the 80% max in order to see size gains or pushing himself hard with a little lower weights 960-80 % max range) and higher reps will do just fine? Or is it all about progressive overload?
It's all about progressive overload.
Progressive overload of both resistance and food.
It doesn't matter if you're lifting 60% of your max or 80% of your max right now, as long as you are lifting MORE next time.
Don't stick to any one weight or rep range. You can benefit anywhere between 3-12 reps of most exercises, some up to 15+ on legs. Train with something for a while, enough to make progress but not get your body too used to it, then change to something else.