I usually only see brackets like that when I'm working with bookmarks. But they are pretty easy to remove - so much so that other users have accidentally deleted them on me.
A search on the web found this - the suggestion that the file may have restriction on parts of the document to prevent them from being edited. That post is found in the Microsoft forum:
Redirecting
Presumably, if you go to the Review tab on the ribbon and toggle Restrict Editing, it might go away if that's the cause.
Does Reveal Formatting (SHIFT + F1) show you anything? I just tested this on a bookmark, and though I could see the brackets, Reveal Formatting showed me nothing.
If you go under File/Options/Advanced, roughly 2/5 of the way down the Advanced panel there is Show document control. If you have "Show bookmarks" ticked, and Field shading set to "Always" you might try to unselect/change those to see if the problem goes away. It could be a field or cross-reference (gray shading) inside a bookmark [brackets], but I'm thinking it has more to do with restriction.
The weird thing for me is that it prints/converts to PDF with the brackets showing. None of my scenarios does that.