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Korekiyo used a "seesaw effect" by jumping on a loose floorboard in order to impale a hidden under a cage Tenko Chabashira on a knife. He did so despite having already killed Angie Yonaga whose murder lacked enough evidence to point to the culprit. As a result, the term "seesaw effect" was brought up numerous times during the third Class Trial.
Because Korekiyo insisted on using the seesaw trick, people have started making so many memes about it, the natural outcome was for people to jokingly ship him with seesaws. The memes and joke ship were fueled by the fact that Korekiyo got greedy for another kill (due to his goal of making 100 friends for his Sister), even though he would have been uncaught due to lack of proper final evidence had he stopped at his first murder. The fans jokingly say that he got greedy for another kill because he didn't want the trick that he prepared go to waste as well as because of his love for seesaws.
The memes are also spread through some YouTubers' gameplays, with possibly the most famous one being NicoB who is known for his funny playthroughs and even is responsible for creating some Danganronpa memes. The ship may also serve as a sort of coping mechanism for a rather dark and villainous third case of Danganronpa V3.
The memes tend to mock Korekiyo for still going with the seesaw trick and killing Tenko, which made him way more suspicious, instead of stopping at killing Angie, in which case there wouldn't be enough evidence to point him as her killer. Meanwhile, fan fictions tend to explain his irrationality with a childish love for seesaws. Some even take the love much further, making him sexually attracted to seesaws.