Working independently means, among other things:
You can be given a less-than-detailed specification and work out the details yourself.
You can solve many problems on your own, whether it’s fixing a bug or figuring out a new part of the code base.
On the other hand, you know when to ask for help and how to ask for help.
You can check your work’s correctness on your own.
You are able to do project management for your own tasks.