This is probably the worst advice that anyone can ever give anyone in the gym. While you definitely don't want to lift things that are too light and you DO need to push your limits, this idea that you have to badly damage your muscles in order to see faster gains is horrific advice that could provide people with long-standing chronic pain.
I understand that it is just a mantra because it rhymes but there are a lot of people that actually do stick to this way of thinking or at least joke about it. There is one guy that I know that says that "he doesn't feel like he did enough at the gym unless he is in pain the following day."
There is a massive difference between struggling to get through the last 2 reps of a curling routine and another to experience a sharp pain your elbow every time you do a curl. I once decided to hulk out on some rotating shrugs in the hope that I would build greater bulk in my neck / shoulders only to end up sidelines with a week and completely incapable of moving my neck and had difficulty sleeping. This was because I didn't read the "fine print" about how you are supposed to reduce the overall weight you are shrugging by 2/3.
This doesn't mean that you shouldn't do anything difficult in the gym but honestly if something hurts, you are probably doing the exercise wrong or are lifting entirely too much weight.
Calum von Moger is ripped enough to have been chosen to play a young Arnold Schwarzenegger in a film and one day, while dicking around in the gym with too much weight, tore a bicep completely off and this resulted in a forced seriously long break from working out entirely.... This is his paycheck by the way.
I've said it over and over again to friends of mine that got inspired by my turning around of my overall fitness and that is that you need to focus much more on form than you do on weight when lifting. It is also important to write things down and gradually increase what you lift. Don't try to be a hero because if you are in pain after, or especially during working out, there is a very good chance you are doing it wrong.
There is no fast-forward button when it comes to fitness. It needs to be a gradual lifestyle change and I think a lot of the media that exists out there doesn't help people to do this correctly. Be patient! Set small, reasonable goals, and keep a journal. You'll get there and the long-game is far more important than the short one. You don't want to be a guy with chronic back problems because you tried to hulk out in the gym on deadlifts.
You can do it!