Answer
According to Quora, Japan annexed Korea from 1910 to 1945. Japanese rule was oppressive for the first decade or so but was less so after protests in 1919, but toward the end of the 1930s during the Sino-Japanese War, a push was made to fully integrate the Koreans into the Japanese Empire by stamping out their ethnic identity, culture, and language. This period of cultural suppression combined with political and social oppression put a bitter taste in the mouths of Koreans toward the Japanese, and as it all happened relatively recently, the bitterness has yet to fade. There also remain certain particularly-inflammatory issues such as the Japanese forcing Koreans into sexual service as comfort women and the Japanese claim on Dokdo, which rankles Koreans because it is another instance of the Japanese trying to take their land--which the Japanese also tried to do in the late 1500s in a failed invasion that Koreans remember quite well--and all these issues are further fuel for the bitterness left over from Japanese rule.
It should also be said that Koreans have a cultural concept called "han," which is a deeply-felt and enduring sense of having been wronged, a sense of sorrow. Han is an integral part of Korean culture and a key part of understanding Korean attitudes and behavior. This bitter sorrow generally manifests itself in a strong motivation to persevere and excel in spite of--and perhaps to spite--one's oppressors as well as a very long memory for offenses and a particular sensitivity to them. In other words, Koreans perceive that they have suffered, feed that pain by continually remembering their suffering, and let it drive them and shape their identity. The current conflict between Korea and Japan is largely the result of Koreans being apt to passionately hold on to past issues and the Japanese tendency to brush inconvenient truths under the rug to avoid shame.
Koreans want their sorrow known, acknowledged, and assuaged, and so they demand apologies and that what they see as their land be returned in full. This puts the Japanese government in the tough spot of complying at the cost of their self-image or trying to ignore or downplay the guilt of previous generations, and the mere fact that Korea puts today's Japanese in this situation and won't let things slide quietly into the past proves irritating in Japan for some. Many Japanese likely do not fully understand the extent of Korea's suffering at the hands of Japan because the government continues to avoid and/or minimize the issue to some extent, so Koreans' anger likely seems unwarranted as if Koreans simply hated Japanese. National and personal pride then come into play on the part of some Japanese as well, and you end up with mutual rancor and a general disagreement about what really happened and how bad it really was.
All that said, many, many Japanese and Koreans still take an interest in the culture of the other country and don't harbor much animosity toward each other. The animosity that is there is mostly concentrated in Korea. Japanese, as far as I have seen, often enjoy Korean pop music and food, and many, at least among the younger generation, express interest in going there.
Hope this helps! <3
