First it melts, and then it burns. What burns is actually melted wax! The heat of the candle melts the nearby wax, and that melted wax is absorbed by the candle's wick. The melted wax burns on the wick -- the wick itself isn't burning, just the wax on it. However, as the candle burns down, what happens is there isn't enough wax nearby to melt, and so the end of the wick does burn a little, which is why it slowly gets shorter (at the same speed as the candle gets shorter).
It is a chemical change because there is new matter created but flammability is more of a property than a chemical change. A physical change is still the same matter but is in a different form like cutting paper. Its still paper. But if you burn paper, that's a chemical change because its new matter now and its really hard to change back