What are the factors behind making ice faster?
The simplest explanation would be that it isn't (pure) water at all. It is most likely a supersaturated solution of sodium acetate - $\ce{CH3COONa}$ (or sometimes sodium thiosulfate - $\ce{Na2S2O3}$) in water.
A supersaturated solution is a solution that contains more solute than normally soluble at those conditions. So as soon as a rough surface is provided for nucleation (formation of crystals), the solute starts crystallizing and comes out of solution as a big crystal.
The rough surface used over here is most probably the rough end of the burnt straw. This produces enough disturbance to cause crystallization. Usually an already prepared crystal of pure sodium acetate is used for nucleation but only if you want high purity crystals.
This is a pretty common trick (also the original video is from trick life -- as mentioned on the youtube description --) and most people who don't know about this phenomenon are usually easily tricked (then again, the &quot;ice&quot; formed would actually be hot since it is an exothermic reaction but you can't feel it through a video right :) ).
See this page and this page if you want more info.