How does atan(1) * 4 equal PI?

I'm including this little gif from Wikipedia as a great way to understand radians.

The function $\arctan\colon \mathbb{R}\to (-\frac{\pi}{2},\frac{\pi}{2})$ is the inverse of $\tan$. (for the right domain of definition). As $\tan \frac{\pi}{4} = 1$, this means that $\arctan 1=\frac{\pi}{4}$.

Regarding your question about angles: angles are (in mathematics) measured in radians (in $[0,2\pi)$ or $[-\pi,\pi)$), not in degrees: you should expect a value or order $\pi$ or so, not ranging between $0$ and $360$.

You certainly know that $\sin{\frac{\pi}{4}}=\cos{\frac{\pi}{4}}=\frac{\sqrt{2}}{2}$ so one has $\tan{\frac{\pi}{4}}=1$ and therefore $\pi=4\tan^{-1}{1}$

Math explanation from a non-math person:

In a right angled triangle if the two short sides are equal, the angle is 45 degrees.

45 degrees in radians is π/4. (The full circumference is 2πr, 180 degrees is π and 45 degrees is π/4)

sin π/4 = cos π/4 because the two sides are equal.

tan π/4 = tan 45 = 1.

Arctan(1) is the degree (or radian) which returns a value of 1. So arctan of 1 is either 45 degrees or π/4.

π = 4*arctan(1)

This shows geometric explanation for relationship between tan, atan and Pi.

Because horizontal segment AB = 1 and vertical segment BD = 1, angle alpha = 45°. From there you can use atan( BD ) to determine 45° in radiant and take that times 4 to get Pi.