By the central limit theorem applied to the i.i.d. score contributions: since $\mathbb{E}_{\theta_0}[\nabla_\theta \log f(X, \theta_0)] = 0$ and $\operatorname{Cov}_{\theta_0}(\nabla_\theta \log f(X, \theta_0)) = I(\theta_0)$, \begin{align*} \frac{1}{\sqrt{n}} S_n(\theta_0) = \frac{1}{\sqrt{n}} \sum_{i=1}^n \nabla_\theta \log f(X_i, \theta_0) \xrightarrow{d} N(0, I(\theta_0)). \end{align*} Thus $I(\theta_0)^{-1/2} \cdot \frac{1}{\sqrt{n}} S_n(\theta_0) \xrightarrow{d} N(0, I_p)$, and \begin{align*} T_n(\theta_0) = \left(I(\theta_0)^{-1/2} \cdot \frac{S_n(\theta_0)}{\sqrt{n}}\right)^\top \left(I(\theta_0)^{-1/2} \cdot \frac{S_n(\theta_0)}{\sqrt{n}}\right) \xrightarrow{d} \chi^2_p, \end{align*} as the squared norm of a standard $p$-dimensional normal vector.