How can they tell that

When you state out loud that you think you may have injured yourself, or if you show outward signs of such (e.g. clutching your chest during cardio) or if you're sparring too heavy, mainly. Ofcourse trainers aren't psychics and anyone can be ignoring his safety signals and concealing it, but in that case the trainers could hardly praise them for doing it any more than they could be telling them off.

how is that even possible that there is not for ever 1 such case a 1000 cases of people just being lazy or weak-willed and over-react

Well everyone is there of his own free will and his for his own sake. If you under-train you're only cheating youself, so there's an incentive not to be a molly. And the fact that you bother to show up to training means it's probably an effective incentive - at least it was for me.