When the sun sits low, its light travels through far more atmosphere to reach you, at the horizon, roughly 40 times the path length of overhead noon sun. Air molecules scatter short wavelengths (blue) much more strongly than long ones, so by the time that light arrives, the blue has been scattered away and what remains is warm: gold, amber, red.
The same physics makes the sky blue at midday and the sun red at sunset. It's one phenomenon viewed from two angles.
The windows
Duration depends on latitude and season, because it's governed by the angle at which the sun descends. Near the equator the sun drops nearly vertically and golden hour can last barely 30 minutes; at high latitudes in summer it can stretch for hours.
Scattered or broken mid and high clouds: the best case. Undersides catch color, gaps let the low sun through.
Clear sky: clean and harsh, with the last twenty minutes doing all the work.
Overcast: the low sun is blocked before it can light anything. Flat.
Clearing from the west after rain: the jackpot: clean air, dramatic remnant cloud, and an open horizon for the sun to slip under.
What visibility tells you
Haze, smoke, and high humidity mute color and reduce contrast, light gets scattered into a general glow instead of picking out edges. A station reporting 10 miles visibility after a frontal passage means clean dry air, which produces the most saturated color. Low visibility with haze produces soft, washed light that can be beautiful in its own way but won't be vivid.
The often-better half
Blue hour is quietly the more useful window for anything involving artificial light. Sky brightness falls to roughly the level of streetlights and windows, so buildings and cities can be exposed without blowing highlights or losing the sky. It lasts 20 to 40 minutes at mid latitudes, and most people have packed up before it starts.