Per my Turkish husband, this is the same as saying, "Bahçede hiç bir şey yok" although I think the translation would be slightly different. The herhangi version is literally there is not anything in the garden, while the hiç version is literally there is nothing in the garden. Personally, I think in both English and Turkish you would pretty much always use the hiç version. It just seems much more natural.
For the most part I'm just as confused but I think I kind of understand this one.
Herhangi bir şey means anything, and hiçbir şey means nothing.
But if used with yok, both effectively translate to nothing.
So the example given literally translates to "There is not anything in the garden.
But if we replace hernagi bir şey with hiçbir şey, it would literally mean "There is not nothing in the garden," which is incorrect English but means the same thing.
Hopefully that kind of helps, but this section is super confusing, I agree.