It's about the National Health Insurance system. For a 'family' in Korea, the entire family members can get national insurance service if 'at least' one person in family works and pays for the insurance. However, for homosexual couples, it's not the case because they are not considered as 'families'. Now the court ordered so that the insurance policy is also applied to homosexual couples. It means, even if only one person is working and paying the insurance fee, another one (probably not working and not earning money) can get insurance benefit from this.

Well, that's what some people call 'benefits'

As I understood it, the court is essentially saying there needs to be a civil union (though they might not call it that) option akin to what a lot of countries passed before gay marriage got enough public support. And it is usually easier to go from civil union to full equality in marriage, because you can point to those relationships and say "hey, they basically have marriage in all but name already and it is hurting no one".