What traits make mountain goats so well adapted to mountain living?
Mountain goats belong to the subfamily Caprinae, or goat-antelopes. The higher family is the Bovidae (cloven-hoofed mammals). 
Now members of Caprinae are pretty stocky in their build and have thicker fur both of which help them survive in the higher altitude. Their ability to scale/descend sheer cliffs with such agility comes down to their cloven hooves. Their hooves enable them to grasp onto the the face of the mountain in areas other animals could not as well as shift direction very quickly. Despite the fact that cloven-hooves are not unique to mountain goats or other members of Caprinae it is a trait essential for them to be able to inhabit those harsh environments. 

Nubian Ibexes (also a member of subfamily Caprinae) are experts at climbing as well; one of the best Caprinae members in my opinion. Nubian Ibex baby and fox