Nearly every outdoor shoe has this sling on its heel. No matter whether it's a hiking-boot, snowboard-boot or something else.
What's the purpose of it?
The main use is pulling your shoes on, this is particularly obvious in rock climbing shoes that will often have multiple loops so you can really yank on them. Alternatively this can be used as an attachment point.
From James Jenkins in the comments below - en.wikipedia.org/wiki/Bootstrapping "Tall boots may have a tab, loop or handle at the top known as a bootstrap, allowing one to use fingers or a boot hook tool to help pulling the boots on"
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.
They can also be held as a stabilizer while you put the shoes on, but without tugging.
My podiatrist and orthopedist recommend not using it as a tool to pull on the footwear, even if it's high quality. After a year or two, especially if used frequently, that's enough to alter the shape. That causes breakdown of the inner structure so you lose proper support of the feet, heels and ankles. Eventually the legs and back can be affected, as they're relying on your feet for strength and proper posture.
Also, shoes with a compromised shape may have a shorter life, which we all like to avoid, especially since they're so expensive!
I've been instructed to unlace enough of whatever type of shoe I use so my foot can slip in easily. Holding that tag is a great way to keep it in place during that process.
Already some great answers here:
Along with the purpose of tieing them to a sling, and mainly for pulling the shoes while putting on or taking them off, I use them as an additional loop to pass the excess shoelace. This only makes sense with a shoe with high anklets.
A shoe like the one shown above, usually come with a longer shoelace. I usually, leave alone the upper two Eyelets and instead encircle the laces around the shoe, passing them through the sling you are referring to and tie a knot in front.
This ensures that my leg/shin don't rub strongly against the hard topline of the shoe, specially when I am descending. By passing the through the sling makes sure that the laces don't shift up and get loosen up.
But I agree, the most useful feature of those slings is for pulling the shoes and tieing them up when needed.