한국 사람들은 = Korean people (subject phrase). 떡국을 먹어야 나이를 한 살 먹는다고 = that you need to eat rice cake soup to gain a year (object clause). * 떡국을 먹어야 = 떡국을 먹다 + -야(만) (strong conditional) = only if you eat 떡국. * 나이를 한 살 먹는다고 = 나이를 한 실 먹는다 + -고 (quote suffix) = that you gain one year in age. (나이를 먹는다 means 'to gain in age; grow old'. 먹는다 has a few different usages). 생각해 왔습니다 = have thought until now (verb phrase indicating a continuous practice until now). So literally, "Korean people have traditionally thought that you must eat 떡국 on the New Year to get a year older. Similar sentences: - 사람은 먹어야(만) 산다 = People must eat to live. - 어떤 사람들은 하루에 정확히 세 번 밥을 먹어야 건강에 제일 좋다고 주장해 왔다 = Some have been arguing that having exactly three meals a day is the best for one's health.