How can I analyze a quote from The Crucible effectively and in a discerning manner while also analyzing the aesthetic features that show key...

In order to analyze a quote effectively, it helps to know what point you are trying to make. Then, you would explain what the quote means, as well as how it supports whatever your claim is. The quote you provided is lengthy, so you might begin by paraphrasing its most important aspects: Abigail threatens the other girls, implying she has seen violent murders committed and has no problem viciously murdering the girls if they tell on her. 

After explaining the most important parts of the quotation, you would then need to connect it, via your own explanation, to your topic sentence or thesis statement. For example, if you were to argue Abigail is to blame for the witch trials, you might analyze this quote by saying it proves Abigail actively hides facts from authorities and refuses to take responsibility for her own actions, actions that may have made Betty and Ruth very ill and could jeopardize her uncle's position. Abigail's willingness to resort to violence against her "friends" proves her capable of committing violence against people she doesn't value. You might say this quote proves Abigail understands she has done things that would be considered wrong by her community and that this makes it seem as though she will continue to do wrong things whenever it suits her.

In terms of the quote's "aesthetics," Abigail begins by using very short, quick, sentences: "Now look you. All of you. We danced. And Tituba conjured Ruth Putnam’s dead sisters. And that is all. And mark this." This variation in syntax — normal sentence structure — shows she is quite serious, even threatening. If you read those lines aloud and really pause at each period, you can hear just how measured and frightening Abigail sounds. The next two sentences are quite long:

Let either of you breathe a word, or the edge of a word, about the other things, and I will come to you in the black of some terrible night and I will bring a pointy reckoning that will shudder you. And you know I can do it; I saw Indians smash my dear parents’ heads on the pillow next to mine, and I have seen some reddish work done at night, and I can make you wish you had never seen the sun go down!

All of the "ands" seem to compound the terrible things that Abigail has seen and that she is willing to do to keep the girls quiet, including, "and I will come to you," "and I will bring a pointy reckoning," "and I have seen some reddish work," and "and I can make you wish." After her first several clipped, quiet, measured sentences, her sentences get longer and more threatening. This is one way in which you might analyze her sentence structure; it seems to show how comfortable Abigail is with manipulation, how strongly she feels about getting what she wants, and how far she will go to get what she wants.