How does the "noble lie" help mankind in Plato's Republic?
One of the things that Plato is concerned with in this book is creating a perfect and stable society. The noble lie (sometimes called the "magnificent myth") is meant to help people be content with their place in the society. If they are content, the society will be stable.
The noble lie would explain to the people why there is hierarchy in the society. It would tell them that various people have different elements (gold, silver, bronze) in them and that they are, therefore, naturally meant to be stratified. This would not be true, but if people believed it, it would be helpful. The people would not wonder why the hierarchy existed. They would not be angry at their place in that hierarchy. Instead, they would accept their place in society as a natural and proper thing.
Looking at it this way, you can see how the noble lie helps mankind. It helps keep people contented in their place in society. By doing so, it helps them be happy and it helps to ensure that their society will remain stable and free from rebellion by disgruntled people.