How does sulfuric acid catalyze the reaction of potassium chlorate and sucrose?
I think the sulfuric acid isn't really a catalyst here -- I think you use the heat from the reaction between sulfuric acid and sucrose (which will start happily at room temperature) to start the reaction between the potassium chlorate and the sucrose (which usually won't).

ETA: Once you've done that, it's pretty much a straight-up messy combustion reaction, only with the oxygen being provided from a solid source (thermal decomposition of chlorate to chloride and oxygen as the reaction you mentioned shows) rather than anything mechanistically clean. Compare, for example, this fairly typical demonstration just using hot chlorate and a sugar source: Oxidation of Sugar or Gummi Bear with Potassium Chlorate.