What do people mean when they say "no pain, no gain" about exercise?

Just as it sounds.: When it comes to weight training, it's felt that you need to work out to the point that you're sore for a day or two thereafter to make meaningful gains. Not necessarily true, but generally felt to be the case.

Old adage: Which in essence means that if you want exercise to really benefit you, understand that you may have to suffer with some discomfort during and shortly after exercising. As long as your are healthy this is for the most part true, though you shouldn't be in severe discomfort with anything. If you have an injury or chronic medical problem; this doesn't necessarily apply!

Muscle stress hurts: When you work out, for example, doing a modest bench press, the goal is to get stronger. Our muscles respond to appropriate stress by strengthening over time. Appropriate stress can be moderately painful. Hence the term, "no pain, no gain".