How is money put into the economy? ie how is wealth created?

Most answers to questions like mine beg the question of where the money actually comes from, how it actually enters the economy, and to whom is it actually distributed when more money needs to come into the system. This ties into the creation of wealth. No matter how big the system, 3 people or 3 billion people, it seems impossible to create wealth without creating more money. Otherwise you are just shuffling money around, but not creating new wealth. For example: Suppose there are 3 people, who have basic needs met (food, shelter), living on a desert island. They each have 10 $1 bills. Joe creates widget from coconuts, and sells it to Tom for $5. Tom makes a necklace and sells it to Bill for $5, and Bill creates a cup and sells it to Joe for $5. Although three new products have been added to this small (but growing!) economy, there is no creation of wealth without the creation of more money. (assuming you can't barter)
So for the economy to grow - they must agree to increase the money supply to reflect this growth. Now I know the federal reserve prints money, but how do they get it into the economy?