A bruised reed is worth nothing. It will not make a flute, it will not bear weight, it cannot even be leaned on. The sensible thing is to snap it and throw it away. Smoking flax is the same — a wick that has gone out, giving off smoke and no light, useful to no one. Any reasonable person clears both away and starts again with better material.
Isaiah says the Servant of the Lord will do neither. A bruised reed shall he not break, and the smoking flax shall he not quench. He will not finish off what is already half-ruined. He will not put out the last faint sign of life because it has stopped being impressive.
This matters most for people who have quietly written themselves off. There is a particular loneliness in feeling that you were once useful and are not anymore — that your faith used to burn and now only smokes. The temptation is to assume God has moved on to sturdier material, and to keep your distance until you have something better to bring.
The verse forecloses that. The Servant's gentleness is not weakness or low standards; the same passage says he shall bring forth judgment unto truth and will not fail. He is strong enough to be tender. He does not need you at full strength before he will handle you at all.