Masa, 1) The author chose the order of the words for reasons of style and emphasis. If you change the order it will have approximately the same meaning but not the exact meaning the author intended. 2) "The" is used as an adverb in this case. It is a remnant of Old English grammar (around 900 AD) when English had an "Instrumental case". The meaning of instrumental 'the' is "by that, or by so much". Before a comparative adverb or adjective "the" forms an adverbial phrase modifying the predicate. .... that it might burn the quicker (by that much quicker ) she lighted it with a handful of straw. It will burn quicker with the straw than without it. Examples If you plant your corn earlier, it will come up the sooner. = If you plant your corn earlier, it will come up by so much sooner. He moved his chair to the window the better to see outside. = He move his chair to the window to see outside 'by that much' better. He will see outside better than he would have, if he hadn't moved the chair. The sooner the better. = By that much sooner, by that much better.

In this case, "the quicker" may be a bit of an older usage, to my knowledge. In modern English, you'd most likely say that she lit the fire with a handful of straw so that it would burn faster.