To be honest, I have never in my life paid any attention to this stuff (except for its sieving application when reading postdoc job applications). I read papers because of the title/abstract, or knowing what the content is about. I pick a couple when forced to by journals, and didn't think anyone actually cares. It's like the "Keyword" stuff I am forced to create (which again I never look at for papers I read). Are those actually used for something?

I agree that for NSF logistics and job applications it is useful. But the question is focused on its role in research papers, and that is where I don't get it, and is what my puzzlement is about. I never look for papers in my areas of interest by searching for MSC numbers, and I never look for papers by searching for a "Keyword". I am really baffled by the tradition of putting these things in research papers.