Given a pair of RVs $z,\theta$ governed by a joint distribution $p(z,\theta)$. Conditional expectation of $z$ given $\theta$ defines a deterministic function (called as regression functions) $f(\theta)$ given as
$$ f(\theta) = E[z|\theta]=\int zp(z|\theta)dz $$
We have to find the root $\theta^*$ at which $f(\theta^*)=0$.
Now, from the definition of maximum likelihood solution, MLS $\theta_{ML}$ is a stationary point of the log likelihood function and hence satisfies
$$ \frac{\partial}{\partial \theta}\bigg\{\frac{1}{N}\sum_{n=1}^N \ln p(x_n|\theta)\bigg\} = 0 $$
Exchanging the derivative and the summation and taking the limit $N \to \infty$, we have
$$ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N \frac{\partial}{\partial \theta}\ln p(x_n|\theta) = E\bigg[ \frac{\partial}{\partial \theta}\ln p(x|\theta) \bigg] = 0 $$
Hence, we can conclude that finding the maximum likelihood solution corresponds to finding the root of a regression function. How can we reach at this conclusion?
For details, refer Pattern Recognition - Bishop Section 2.3.5 Page 95