It works roughly like this: Patient number #xxx, diagnosed today. Last Monday, he used a taxi (11:30-12:00) to visit McDonalds next to Hongdae station (12:00-12:50) and then walked to Woori bank nearby (1:30-2:00) and used another taxi to go home. Next Tuesday... blah blah...
And then you find out the taxi drivers, all the McDonalds workers, and everybody who used McDonalds and the bank at the same time, and tell them to go self-quarantine for two weeks. And also test them. The idea is basically catch all the virus before it has a chance to run into more people.
It does help that Koreans carry cell phones and use credit cards everywhere. (And, yes it does raise the question about privacy - which is something we should revisit once the emergency is over.)
History teaches that once you give up rights in time of emergency, it is very, very difficult to get them back. Americans are still not back to their pre-9/11 rights to privacy and at this point will likely never will be. Coronavirus will eventually fade away (even if it's a long, bloody road to get to that point), but the loss of rights will persist for generations.
That's an American thinking. The Korean thinking is that we never gave up our rights - our country was born without one, liberated by greater powers, split into half, ruled by dictators. We won our rights with fight, and because it's ours, and our government serves us, we can let them use our power when the society needs it, like a master trusting the key to its faithful servant.
And if they do not relinquish it later, no problem, we'll fight and get them back again.
And then you find out the taxi drivers, all the McDonalds workers, and everybody who used McDonalds and the bank at the same time, and tell them to go self-quarantine for two weeks. And also test them. The idea is basically catch all the virus before it has a chance to run into more people.
It does help that Koreans carry cell phones and use credit cards everywhere. (And, yes it does raise the question about privacy - which is something we should revisit once the emergency is over.)