Let's look at it step by step:

$$1/\sigma^2_n=-\ell''(\hat\theta_n)$$ is true by the definition of $\sigma^2_n$: we are approximating the posterior by a $N(\mu_n,\sigma^2_n)$ distribution.

$$-\ell''(\hat\theta_n)=\sum_i -\ell_i''(\hat\theta_n)$$ is just the definition of $\ell$ and $\ell_i$

$$\sum_i -\ell_i''(\hat\theta_n)=n\frac{1}{n}-\ell_i''(\hat\theta_n)$$ is algebra

$$n\frac{1}{n}-\ell_i''(\hat\theta_n)\approx n\mathbf{E}_\theta[\ell_i''(\hat\theta_n)]$$ is the Law of Large Numbers applied to the independent and identically distributed $\ell''_i(\hat\theta_n)$. This is a slight abuse of notation: up to now we can think of $\hat\theta_n$ as a complicated function of the data, and so a random variable whose distribution depends on $\theta$ and whose value depends on all the $\ell_i$. On the right-hand side now, though, we mean the number that is the value of $\hat\theta_n$ in the current dataset. Suppose $\hat\theta_n=7.3$. The right-hand side is $n\mathbf{E}_\theta[\ell_i''(7.3)]$

And finally $$n\mathbf{E}_\theta[\ell_i''(\hat\theta_n)]=nI(\hat\theta_n)$$ is the definition of Fisher information (again with $\hat\theta_n=7.3$ or whatever).

The reason it's Mostly Harmless to switch between the two versions of $\hat\theta_n$ is that the limits are uniform in $\theta$ for all $\theta$ close enough to the true value: the fact that $\hat\theta_n$ is a function of the $\ell_i$ doesn't actually matter in smooth parametric models. On the other hand, the fact that it doesn't matter is a bit tricky to prove, and is the hard part of a lot of basic asymptotic results: you can often derive an expansion that's what you want, but showing the remainder is small uniformly in the parameter is the hard part.