Can you describe the science behind home smelting (Details inside)?

Not knowing the exact design of the forge, I would assume it is the same basic standard as a bunsen burner. When the propane is turned on low, it is not operating efficiently and you are getting incomplete combustion which shows as large yellow or red flames. When you turn the gas up, it increases the efficiency (based on flow designs of the torch/burner/forge) and the flame seems to shrink but is actually getting hotter. With a Bunsen burner, you can see the flame turn from yellow to blue which indicates complete combustion. In the video you provided, I suspect it is heating up the ceramic lining inside the forge which gives it that orange glow.

The velocity of the jet of flame increases.