Given that about 1% or less of my working day will be concerned with the details of the design/development methodology in use, how important is it for someone coming from a traditional waterfall development model environment to have had experience with agile methodologies?
It is important, mostly because your "given" is wrong by a factor of 10+. Agile is a lot about you doing work. It's no longer the boss telling you exactly what and how to do. It's you, planning, estimating, communicating, collaborating. That takes time. It requires certain traits that are not required and actually despised by most traditional project managers. And it follows certain rules. Rules you don't know (yet).
If the answer is that it is important, then what aspect(s) of Agile are at issue here that they require prior experience?
Everything? I mean look at your programming skills. Would you say "why would I need experience in that language"? Experience means you don't have to be trained, you can adjust faster, hit the ground running, whatever your favorite term for it is. You will also know if it is for you. Maybe you won't like Agile? It's better to have a candidate who did it and liked it instead of a "maybe". There is no such thing in Agile as leaning back, letting the other folks do it and tolerate it. You are in... or not.
Your best bet is to get a book and familiarize yourself with Agile. Most companies I know will accept somebody who says "I always wanted to do that, I read about it and heard about it on conferences, but my old company won't do it". That's fine. It's really not up to you do decide what your current company uses, and your potential employer knows that. However if you interview and say wildly uninformed things like your first sentence here, that will probably mean they will go with a candidate who, even though they have zero experience like yourself, have at least some book knowledge.
"Not having experience in Agile" where I live also means something else: that person has never visited a conference. Because you cannot escape it. Whether you like it or not, whether your company does it or not, you will have heard of it, you will have had multiple chances to have workshops and talks about it, people will give you cards to free online courses... there is no way to not at least get basic knowledge, except you actively chose to not get informed. And personally, I would doubt a persons ability to stay on top of software development if they don't take any chance to learn something new.