Translation:I am who I am.
Shouldn't the sentence end with something other than a ~다? Eg. Conjugation or ~예요 or ~이에요?
Yes and no.
While talking you usually end your sentences with -아/어(요) or -ㅂ니다/습니다, but in written text sentences can end in -다, too, when not adressing anyone and just making factual statements like "The school is very old." which would be "학교가 매우 낡다."
This also works for the past like "교실이 매우 작았다." ("The classroom was very small.")
" I am who I am , with all of my faults and my mistakes" -Kim Namjoon(RM) UN
What is the meaning of the verb 나다 here? A contraction of 나 + 이다? Or is this idiomatic?
Sometimes 이다 is contracted to the sole 다 in Korean. This happens when '이다' is attached to a word that ends with a vowel. '우리는 하나다' - 'we are one!' (하나 - numeral 'one').
Note that it can be still conjugated. 'He is a doctor': 그가 의사입니다 or 그가 의삽니다. 'Who are you?': 너는 누구이니? or 너는 누구니? 'My grandpa is a miner': 우리 할아버님께서 광부이세요 or 광부세요.
In above examples, 이 is just thrown out. The meaning doesn't change.