The golden angle is related to the golden ratio, but it is not as well known. And the relationship is not quite what you might think at first.
The golden ratio φ is (1 + √5)/2. A golden rectangle is one in which the ratio of the longer side to the shorter side is φ. Credit cards, for example, are typically golden rectangles.
You might guess that a golden angle is 1/φ of a circle, but it’s actually 1/φ2 of a circle. Let a be the length of an arc cut out of a circle by a golden angle and b be the length of its complement. Then by definition the ratio of b to a is φ. In other words, the golden angle is defined in terms of the ratio of its complementary arc, not of the entire circle. 
The video below has many references to the golden angle. It says that the golden angle is 137.5 degrees, which is fine given the context of a popular video. But this doesn’t explain where the angle comes from or give its exact value of 360/φ2 degrees.
 Why does this work out to 1/φ2? The ratio b/a equals φ, by definition. So the ratio of a to the whole circle is
a/(a + b) = a/(a + φa) = 1/(1 + φ) = 1/φ2
since φ satisfies the quadratic equation 1 + φ = φ2.