Requires macOS ≥ 10.9
Due to sandboxing requirements, Docs cannot just add files to Xcode's directory, you have to give it permission first. Usually, this is just a matter of granting access to the folder the app pre-selects, but sometimes, it doesn't exist yet. This can happen e.g. if Xcode is newly installed and didn't have a chance to download any updates to its own docsets yet.
If this happens, you need to create the folder manually, by running the following shell command from Terminal:
mkdir -p ~/Library/Developer/Shared/Documentation/DocSets
The simplest way is directly from CocoaDocs — hover over the share icon in the top right corner and add to Docs for Xcode. You can also add a feed manually or just open a docset file.