Take Cauchy's formula on a circle centered at $a$: it says $f(a)$ is the average of $f$ over the circle. An average can never be bigger than every value it averages, and that simple observation controls where $|f|$ can be largest.
Key idea
Maximum modulus principle. If $f$ is analytic and not constant on a bounded region $D$ and continuous on its closure, then $|f|$ reaches its maximum on the boundary of $D$, never at an interior point.
If moreover $f$ has no zeros in $D$, the minimum of $|f|$ is on the boundary too (apply the principle to $\frac{1}{f}$).
If moreover $f$ has no zeros in $D$, the minimum of $|f|$ is on the boundary too (apply the principle to $\frac{1}{f}$).
Why it works
With $z = a + re^{it}$ in Cauchy's formula, $dz = ire^{it}dt$ and $z - a = re^{it}$, so $f(a) = \frac{1}{2\pi}\int_0^{2\pi} f(a + re^{it})\,dt$: the mean value property. Suppose $|f|$ had its largest value $M$ at an interior point $a$. Then $M = |f(a)| \le \frac{1}{2\pi}\int |f(a + re^{it})|\,dt \le M$, so equality holds and $|f| = M$ on every small circle around $a$. Then $|f|$ is constant near $a$, which (by the Cauchy–Riemann equations) forces $f$ to be constant.
Worked examples
Worked example: A linear function
Find the maximum of $|z + 3 - 4i|$ on $|z| \le 2$ and where it happens.
- The maximum is on the circle $|z| = 2$.
- $|z + (3 - 4i)| \le |z| + |3 - 4i| = 2 + 5 = 7$, with equality when $z$ points the same way as $3 - 4i$.
- $z = 2\cdot\frac{3 - 4i}{5} = \frac{6}{5} - \frac{8}{5}i$, where the value is $7$.
Answer: $7$, at $\frac{6}{5} - \frac{8}{5}i$
Worked example: An exponential
Find the maximum of $|e^{iz}|$ on $|z| \le 1$.
- $|e^{iz}| = e^{\operatorname{Re}(iz)} = e^{-y}$.
- On the closed disk, $-y$ is largest at $z = -i$, where $-y = 1$.
- The maximum is $e$, at $z = -i$, on the boundary.
Answer: $e$, at $-i$
Worked example: A minimum
Find the minimum of $|z + 3|$ on $|z| \le 1$.
- $z + 3$ has its only zero at $-3$, outside the disk, so the minimum is on the boundary.
- $|z + 3| \ge 3 - |z| = 2$, reached at $z = -1$.
Answer: $2$
The same idea works for the real part: a harmonic function $u$ (such as $\operatorname{Re} f$) takes its maximum and minimum over a closed bounded region on the boundary. In heat flow this says the steady temperature inside a plate is never hotter than the hottest point of its edge.
Common mistake
Hunting for the maximum at a critical point inside. In multivariable calculus you set the gradient to zero and test interior points. For $|f|$ with $f$ analytic, interior critical points are never maxima (and $f'(z_0) = 0$ is a different thing altogether). Go straight to the boundary: parametrize it, or use the triangle inequality and check when equality holds.