Why does NTM need to derive certificate to prove "If a language is in in NP iff it is decidable by some nondeterministic polynomial time TM"?
By definition NP is the class of languages for which you can efficiently verify with a certificate that a string is in the language.

Informally; it is the class of problems that you can verify the solution to given a witness, but you cannot find one easily. If you could, then it would be in P.

A NTM can be seen as a unrealistic Turing machine which always guesses the right answer. Since the NTM is non-deterministic given an input x. We cannot be sure exactly what path he took to derive the solution, therefore a witness is needed. 

For example;

Problem: Is 24 prime? 

A NTM would guess the answer to be no and the certificate would be the number “4”.

A verifier presented with the certificate can divide 24 by 4 and also come to the conclusion that 24 is not prime.