The heat of the flame causes flammable gasses to be released by whatever is burning, for example vapourised candle wax. This gas is then burned and forms the flame itself. A strong enough wind physically disperses the gas before it can burn, eliminating the flame. A big fire like a campfire will have enough heat to reignite the gasses after a brief gust, whereas a candle does not typically have enough heat.
There is a neat trick where you can relight a candle immediately after blowing it out by touching a match to the stream of smoke; this smoke contains non-ignited flammable gas that is still coming off the wick.
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