How do I go camping in Korea but I do not own any gear and I am not sure how it works here?

Plenty of campgrounds and glamping sites rent or fully provide gear, so you can just show up; search 글램핑 or 오토캠핑 on Naver to find them. Auto-campsites (오토캠핑장) and national park campgrounds need reservations, which you book on the park's site, while gear can be rented from camping shops. If you own nothing, start with glamping, which gives you a set-up tent, bedding, and a barbecue.