When you have a free hand, it means you have permission to do whatever you want. When you have a free hand in the kitchen, you might prepare a gourmet meal or just a bowl of cereal.
There are several phrases you can use to mean "freedom," including free hand, free rein, and blank check. When you are able to make your own decisions and act in the way that seems best to you, you have a free hand. A teacher with a free hand is likely to have classes that are more fun and interesting than one with a set curriculum.