Wicked flame sources such as candles depend on convection to operate. Once lit, gasses in the candle's flame rise and cooler oxygenated air rushes in from the sides and below to replace it. As this new air approaches the flame it rapidly heats up, reacts with the fuel in the wick, ignites and rises, continuing the cycle. This is why the flame's colour transitions from blue (initial ignition) through orange and finally yellow (where the last of the evaporated fuel is consumed). This is also why a candle flame extends up from the wick.

Rapid air movement from blowing or clapping drastically increases the amount of air flowing past the wick, which elongates the flame and lowers the concentration of heat around the wick. Less heat near the wick reduces the rate at which its fuel vapourizes, which makes less of it available to react with the air.

The air itself absorbs heat. Slow moving air absorbs more heat per unit volume, so it more readily reaches the fuel's ignition temperature and sustains the reaction. Faster moving air absorbs less heat per unit volume, making ignition more difficult.

This is a long-winded way of explaining how blowing or clapping your hands cools the area around the wick to the point that ignition ceases.

A slight increase in airflow velocity beyond what simple convection produces actually improves ignition by making more oxygen available at the point of ignition. This is why blowing on a fire causes the embers to glow brighter and increase in temperature. But there's a tipping point beyond which the heat drawn away from the seat of the flame overcomes the more efficient ignition due to extra oxygen from the additional airflow.