What are those black fibers in my mango?
It's an indication that the fruit has started to rot:

When dark spots start appearing on the side of a mango, it's starting to rot. A mango can rot from the pit, the skin, or from the non-stem end. Any black fibers also indicate that the fruit has started to rot, and at that point, you need to toss it into the trash.
From: Student's Vegetarian Cookbook For Dummies, page 301

Edit: Yes, the above is probably being overcautious in stating “any black fibers? → trash!” It's from a “Dummies” book after all. ;) I couldn't really find another good reference. But I'm sure you can make the distinction between a piece of fruit that has a small brown spot and one that's black and shriveled. Still, the color of the fibers is a result of the late ripening/rotting process of the mango, rather than a result of an earlier stage in its growth. But having eaten this mango with the one, tiny black fiber shown in her picture, I don't really expect to read Mien's obituary on this site tomorrow. ;)