Edit: For clarity, I am not asking how to turn the sidebar on.
Not sure what to call them but this is what I mean:
In red, is what I'm calling a "folder guide". Just a nice line to help your eye better visually track the hierarchy of folders.
I'm using the Cobalt2 theme and haven't looked into alternate themes yet but it would be nice if it was possible to add something like this without changing to a new theme.
It's not possible (at least at the time of this writing) to have hierarchy lines in the tree view of the side bar such as in your image above. There is an open issue in the unofficial issue tracker for it, however.
That issue has been open for a while, but prior to the recent-ish release of Sublime Text 3 a lot of work went into the sidebar behind the scenes, so possibly something like this might be added in a future update.