Intuition Behind Maximum Principle (Complex Analysis)

Think about what the mean value property of analytic functions says: $f(z_0) = \dfrac{1}{2\pi} \int_0^{2\pi} f(z_0 + re^{i\theta}) d\theta$, where $f$ is analytic in the disk $B_r(z_0)$. This says that $f$ is equal to the average of the boundary points. How can $|f(z_0)|$ be larger than every single point of which it is the average? Thinking discretely, if $a= \dfrac{a_1 + \cdots + a_n}{n}$, can $a$ be larger than every point in this sum? No, this is not possible.

holomorphic means, among other things, that the map is open. This is immediate when $f'(z_0) \neq 0,$ the Inverse Function Theorem says that an open neighborhood of $f(z_0)$ is covered surjectively. Even when $f'(z_0) = 0,$ the surjective part still holds, it is just that the map is locally $k$ to one, where $k$ is the first derivative such that $f^{(k)} (z_0) \neq 0.$ So, there it acts the way $z^3$ acts around the origin, for example.

Anyway, your $D$ is open; assume that the modulus takes its maximum at some point $z_0 \in D.$ Well then $f$ maps a neighborhood of $z_0$ onto a neighborhood of $f(z_0),$ including points with larger modulus than $f(z_0)$