The posters read: "It's safe and convenient: Walk on the right!" and "It's the walking culture of Korea and the world."
Of course, Koreans have always driven on the right side, and usually they walk any old way they want, but until now the general rule has been to walk on the left. This was apparently put into law over 80 years ago, during the Japanese occupation, and I can't help but think that this reversal is something of a subconscious rejection of the Japanese cultural legacy in Korea. Walking on the left is an aspect of the Japanese shinto religion, as shrine visitors are supposed to always keep to the left side when walking around the grounds (I believe this is also true of Zen Buddhism).
This is all just speculation on my part, however. Seoul City gives no such historical reasons. It promotes the campaign with the reasoning that "most Koreans are right-handed, and walking right is more convenient for those people." An article in the Korea Times also provides the following impressive statistics:
The Korea Transport Institute says psychological burden will be reduced 13-18 percent when walk right. It also expects the switch to increase walking speed 1.2-1.7 fold, reduce pedestrian collisions 7-24 percent and pedestrian density 19-58 percent.