Set $q(X, \theta) = \nabla_\theta \log f(X, \theta)^\top \nabla_\theta \log f(X, \theta)$, so that $i_n(\theta) = \frac{1}{n}\sum_{i=1}^n q(X_i, \theta)$ and $I(\theta) = \mathbb{E}_{\theta_0}[q(X, \theta)]$. Decompose \begin{align*} \hat{i}_n - I(\theta_0) = \bigl(i_n(\hat{\theta}_\text{MLE}) - I(\hat{\theta}_\text{MLE})\bigr) + \bigl(I(\hat{\theta}_\text{MLE}) - I(\theta_0)\bigr). \end{align*} For the first term, the law of large numbers applied uniformly over $\Theta$ gives $\sup_{\theta \in \Theta} \|i_n(\theta) - I(\theta)\| \xrightarrow{\mathbb{P}_{\theta_0}} 0$, so the first term goes to zero in probability. The second term converges to zero by consistency of $\hat{\theta}_\text{MLE}$ and continuity of $\theta \mapsto I(\theta)$, via the continuous mapping theorem.