Wax itself barely burns in its solid state. However, the heat of the flame melts the wax near the wick, and the wick draws the liquid upward like a sponge.
At the top of the wick, the liquid wax is heated further and evaporates – it's actually this wax vapor that burns, reacting with oxygen from the air. This produces carbon dioxide, water, and the energy we perceive as light and heat.
The flame itself generates enough heat to continuously melt and vaporize new wax – a self-sustaining cycle, as long as wax and oxygen are still available.