Carrying.
The sling makes it easy to attach the shoe/boot to a backpack or something else. (For me, this is the main point.)
For example, you can clip your approach shoes to your climbing harness (or your pack, if you carry one) if you plan to descend after a multi-pitch route instead of abseiling. Or you might want to take off your shoes and switch to a pair of sandals when crossing a stream. Basically anything that involves the need to carry your shoes.
Another point is that sometimes these slings can also make it easier to put on the shoes. That highly depends on the shoe and situation, though.