Given an arbitrary angle $\theta$ which is not a multiple of $\pi$, to say that $n \theta$ and $m \theta$ are close to each other on the circle is to say that $(n-m)\theta$ is close to $0 \bmod 2\pi$, so to say that the fractional part of $(n-m) \frac{\theta}{2\pi}$ is close to $0$. If you want to avoid this, you want to pick a value of $t = \frac{\theta}{2\pi}$ with the property that, for any integer $n-m$, the fractional part of $(n-m)t$ is never too close to $0$.
Phrased another way, for any integer $q$ we never want $qt$ to be too close to another integer $p$. Phrased yet another way, we never want $t$ to be too close to a rational number $\frac{p}{q}$. So we are looking for irrational numbers which are hard to approximate by rationals.
The closest rational approximations (in a suitable sense) to an irrational number can be read off from truncations of its continued fraction
$$t = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + ...}}$$
and in particular are better approximations if the corresponding entries $a_i$ of the continued fraction are large. That means that the best irrational number for the job is the one with the property that each $a_i$ is as small as possible, so
$$1 + \frac{1}{1 + \frac{1}{1 + ...}}$$
which turns out to be precisely the golden ratio. (Why? Because the above number $t$ has the property that $t = 1 + \frac{1}{t}$, or $t^2 = t + 1$, so it is either the golden ratio or its conjugate, and it's greater than $1$ so it must be the golden ratio.)
