It means that the Europeans had believed that the World was theirs, and that they would be able to do what they like with it. They thought that they'd be able to rule the planet and other nations that weren't as powerful as European nations.
When someone tells you that the world is your oyster, what they're saying in a way is that anything is possible for you on this planet. The planet is "yours" in a manner. You're free to travel and go where you want without restraints.