The wax will not burn without a flame to heat it to a gaseous state. The wick lifts the melted wax up into the flame facilitating combustion. The wick burns slowly because oxygen from the surrounding air is depleted in the flame before reaching the wick.
If you blow out the flame oxygen reaches the wick causing it to smolder (the red part in the dark). Without a flame to burn the gases from the smoldering wick, smoke is released into the surrounding air.
If the wick burns too short it can become completely smothered in liquid wax. Deprived of the oxygen in the air the wick cannot smolder.
Candle Burning and Flame