Why does it do that? Whenever I’ve put a candle out, and soon after try to light it again, it lights up before the flame even touches the wick. Even within a few centimeters. What causes this?
My WAG is that there is still enough ambient heat near the wick that adding just a little more will produce enough heat to light the candle.
that’s exactly it, Khadaji
Sorry to be contrary and all, but I think there is another answer based on an actual experiement we did back in College Chemistry. When a hot candle is extinguished, there are still volatile organic hydrocarbon compounds that are evaporating from the hot candle and the wick, which form a small invisible (or visible; some are in the “smoke” that you see after extinguishing) cloud around it, until they are dispersed. These will be produced until the temperature of the wax near the hot end drops enough, and if exposed to flame they will ignite.
To test this, you can blow out a candle, quickly cover the candle with a glass cup to catch the smoke and VOCs, and then turn the cup over, stick a match in it, and if you do it just right you can see a small flash of flame as the VOCs ignite.
So much of a candle flame’s heat is lost as radiant energy I have doubts that a wick hold enough heat after even a couple of seconds to re-ignite with just the proximity of a match. This of course will depend on how much of an ember there is on the wick - I mean, I’ve seen embers grow to re-ignite the wick, although that’s fairly rare.
that’s exactly it, Una. (What the hell do I know? I spent my chem lab time trying to find new ways of blowing shit up) 
Candles ALWAYS burn only the vapor of the fuel. That’s why a wick can lasts hours in a candle, when it’d burn completely in under a minute, if you ignited it without a candle. The heat of the flame vaporizes wax from the wick, and the VAPORS burn, not the liquid/solid wax or wick. In fact, the evaporating wax actually cools and protects the wick.
Once you realize this, there’s no magic to it at all. You’re just lighting the vapors further from the wick, rather than closer. This also explains why a candle burns with a large hollow spherical flame in zero gravity (as you’ve probably seen in space shuttle demo videos
I’m not up to surfing for good explanations right now, but you should be aware that the famous 19th century chemist/physicist Michael Faraday did the definitive experiments exploring how a candle burns, so adding his name to your searches will probably lead you to more illuminating <ahem> results results than simply
Googling “why a candle burns”
Since my WAG seems to be wrong, (SeekingTruth, you made me laugh) I’ve decided to be helpful and provide a good link verifying what the others have said: How Candles Work
When in doubt, consult the Master.