추다 means the verb 'to dance' and 춤 is the noun 'a dance'. 춤 is actually the noun form of 추다, as it is formed by adding the ending (으)ㅁ to the verb stem. I was told that we should say 춤 추다 'to dance a dance' and that 추다 alone is incorrect.
Hence my question: why?
In a similar way, 'to dream' is 꿈 꾸다, which is literally 'to dream a dream'. You can also read sometimes 잠 자다, 'to sleep a sleep', but it doesn't seem to be mandatory ('Good night' is 잘 자요, not 잠 잘 자요).