Use differential form to prove meromorphic function on compact Riemann surface has same zeros and poles

Question

I am reading mine's modular form note, proposition 1.12 states that the sum of residues of a differential form on compact Riemann surface is $0$. Then he states that applies this to $df/f$, then we can prove meromorphic function on compact Riemann surface has same zeros and poles.

I can not follows this, can someone help? I apologize if this question is too elementary.