I have seen a large portion (but not all) of Muay Thai athletes wrapping their ankles. What are the benefits of doing so? Is it for reducing damage incurred from kicking the opponent or preventing twists? How important is it to do so?
I don't know why kickboxers use ankle wraps, but I just happened to come across Reddit's /r/muaythai FAQ, which says they are for general ankle support:
You may see thai fighters wearing what look like ankle braces while training or fighting. These are ankle supports that can help keep prevent a fighter from rolling his or her ankle while training or fighting. Ankle supports are not required to train in Muay Thai.
I cannot vouch for this information.
It is ideal—in any martial science which delivers blows as heavy as those in Muay Thai—to have some support around the joints most frequently used. This is one reason why the ankle is wrapped—akin to having a knee brace. This provides support and reinforcement
The second reason is for protection. That added layer of cloth, or whatever is used to wrap the ankle may end up cushioning some impact and preventing cuts, tears, and sometimes bruises.
It may prevent burns from scraping the ankle or the top of the foot on the canvas when slipping and getting back up. But I highly doubt that's the reason fighters wear them. I think it's more psychological. I remember when I was a young Taekwondo-ka, if I had wraps around my knees and ankles, it just felt like I could kick better. There's no science to support this, but it felt real enough.
It may also be an attempt to make the shins "harder" or to afford a level of padding.