This phrase has been current in English since the mid 15th century. It means that it's better to have a thing of lesser value in reality than it is to have something of greater value in theory.
For example, it's better to work at a modest job to make a living, than to wait for someone to offer you a high-paying one that might never happen.
One origin of the story is that a huntsman knew that he could easily catch at least one bird if he waited by a stream. There were many more birds in the forest, but he could not be sure he would catch any there if he tried.
What you have (a bird in the hand) is worth more than what you might possibly get (two birds in the bush).