This is probably one of the most common things you will hear from some people training jiu-jitsu but I think the idea of it has become confused.
Straight off... you have to use strength in jiu-jitsu. What is strength? A measure of how work your muscles can do. What do you use to move your limbs? Muscles = strength. And to go further, the entire basis of jiu-jitsu technique is to use leverage in order to multiply your strength and maximise your advantage over your opponent.
So what people really need to say is "Don't prevent yourself from learning technique by relying on strength"... I dunno, maybe it's not as catchy though. You should try to make any movement or control of your opponent be as easy as possible for you; if you're straining to do something then you're probably not doing it right. If your muscles are dead after every roll, you're probably using too much strength and not enough technique.
That said... there is nothing wrong with using your strength flat out at times. At the end of the day, jiu-jitsu is about learning to fight. Part of fighting is using your physical attributes and strength is an attribute just as much as speed, flexibility, cardio, long legs or whatever... and how many times have you heard people saying not to use those things to your advantage? Exactly. It makes me laugh when I hear a little guy telling an ultra heavy "Don't use so much strength" and then they hit a blink-of-the-eye armdrag to back take.
Basically, what it comes down to is that any physical attribute can be a bad thing in your training, but a good thing in your ability to fight. So you have to judge those things for yourself...
- when you're training in general try to maximise your use of clean technique and minimise your use of physical attributes.
- make sure you also do use physical attributes at times. You need to learn how to use these well; using an attribute correctly is a technical aspect of jiu-jitsu.
- pick your partners wisely when you want to go flat out. If you're really strong then there is little point in putting it to full use against a rooster weight.
- realise you also need to learn to deal with your opponents having superior physical attributes. It's no use just complaining when someone is really strong... it's a fact of life that some people are massive/strong. You might have to fight one in a competition or in a real fight, so get yourself prepared for it.