Degree of a normalized polynomial counts roots in the unit disk

Let \(f\in\mathbb C[z]\) and assume that \(f\) has no roots on the unit circle. Consider \[ S^1\hookrightarrow\mathbb C\xrightarrow{f}\mathbb C\setminus\{0\}\xrightarrow{\pi}S^1, \] where \(\pi\) is the norm map. Prove that the degree of this composition equals the number of roots of \(f\) inside the unit circle.