When something burns, like a candle for instance, the substance acting as the fuel (e.g. the wax) becomes very hot and vapourises. Because the vapour is hot, and expands, it has a lower density than the surrounding air so it rises upwards. As it rises it mixes with oxygen in the air and the mixture burns. The hot gases produced continue to rise out of the way, and the extra heat produced vapourises more fuel so that the process can continue. But in space, where there is no gravity, there is also no density gradient so the hot gases can't rise. Instead they accumulate in place and starve the flame of a supply of fresh air, so it goes out. The only way to maintain a flame is to artificially blow in oxygen to ensure that there is sufficient to support combustion.
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