What does “Keep Index” option of Git stash dialog in Intellij IDEA do?

This determines if the stash is created with the git --keep-index flag.

From GIT docs:

If the --keep-index option is used, all changes already added to the index are left intact.

So in other words, any files you’ve added to git will also remain outside of the stash when you create it. Whereas normally newly added files would be removed after stashing. (See this question for a more indepth description of where the index is.)