Equivalence between alternative definitions of NP
The definition of $\text{NP}$ in terms of verification can be formally stated as follows:


  A language $L$ is in $\text{NP}$ if there exists a polynomial verifier for $L$ such that for every $x$:
  $$x \in L \iff \exists u: V(x,u) = 1.$$
  
  Here $u$ serves as a certificate.


To prove that the other definition $\text{NP}$ in terms of NTM implies the one above, you should construct such a polynomial verifier $V$ with the desired property. That is, for every $x \in L$, you need to find a certificate $u$ (you are not given a certificate $u$) such that $V(x,u) = 1$. Note that for every $x \in L$, there is a sequence of nondeterministic choices that makes the NTM $M$ accept $x$. This sequence is the certificate we need. What the polynomial verifier $V$ need to do is just to simulate the action of $M$ using these nondeterministic choices.