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 $360$.