Studs on leather armor like that are usually to hold two or more layers of material together so they don't slide around.
Really, you see that sort of thing very rarely in historical examples. Pictures of "studded" armor are usually actually brigandine, where the studs hold small metal plates to the back of the material, which is usually something like velvet instead of leather, but not always.
Sorry, I didn't have a cool stud joke, so I just had to answer the question, instead.