What is the golden ratio?
The golden ratio, or φ, is about 1.618. You get it when you split something in two so that the large part relates to the small part as the whole relates to the large part. Architects and designers have used it for centuries to divide space.
The golden angle: colors that never repeat
Divide the color wheel by φ and you get the golden angle, about 137.5°. Turning your hue by 137.5° again and again puts each new color far from the previous ones. That is why it is used in charts and data visualization, when you need many colors that are easy to tell apart. Note that it gives distinct colors, not necessarily harmonious ones; for an elegant design, use one or two of them as accents.
The 62 / 24 / 14 split
This is the most useful part. Apply φ to how much space each color takes and you get 61.8% for the dominant color, 23.6% for the secondary and 14.6% for the accent. It is very close to the well-known 60-30-10 rule, and it works for the same reason: one color leads, one supports and one stands out.
Is it a myth that the golden ratio makes colors prettier?
Mostly, yes. There is no evidence that picking hues "in golden proportion" makes a palette more beautiful by itself. What does help is the above: spacing hues well when you need many colors, and splitting space around one dominant color. This tool uses φ only where it adds something real.