I don't know what that person meant but, like many in the region, Koreans have family registries (戶籍) that record their family lineage. At least among people I've known who have spoken about theirs, Korean family registry records tend to go back much further than the median east Asian country.
The names of the systems related to this registry are slightly different in Chinese, Japanese and Korean but you can see links to Wikipedia entries for each of them from here:
https://en.wikipedia.org/wiki/%E6%88%B6%E7%B1%8D
> "Maintain the purest racial pedigree"? What does this even mean in actual terms?
It’s usually, as it is in this case, genetically meaningless. Culturally, it means someone picked an arbitrary point in the past, counted up the cohort of people within some group (and a metric for testing in-ness), and then deemed descendents of only that group to be pure, descendents of that group in part to be impure, and people who have no ancestors in that group to be outsiders.
In practice, recordkeeping past a handful of generations gets annoying, so most cultures set this line somewhere within 200 years ago. (The concept correlates with nativeness to varying degrees.)
