This depends on what it is precisely you are writing about, but the biggest fact you need to remember is that quotes should be used as proof of the main points you are trying to make about the texts you are writing about. I always teach my students the following way of writing essays: Point, Proof, Deduction. You make a point, support it with some proof (normally in the form of a quotation from the text) and then give an analysis of that proof, relating it back to your point.
How and where you use quotations depends on the purpose of the essay, but it is evidence. Evidence is just as necessary when writing about a short story as a novel. It does not matter if it's a novel, poem, or short story. The important thing is how you use them.
The post by kplhardison sums it up best. Quotation marks are used to differentiate your own words from other textual evidence that you may use to support your work. Using no quotation marks implies that the textual evidence that you have borrowed is actually your own. Teachers are particularly strict about this, since a lack of quotes can also imply plagiarism--deliberate or not. You may have also noticed that different types of literature uses quotations in the titles (such as short stories and poems); others use italics (novels, anthologies).
I agree with the posting kplhardison offers. Textual evidence is of the utmost importance when writing an essay about a short story, or any story in fact. The essay reader needs to understand from what point-of-view you, as the essayist, is writing, the text in content, and the essayist's interpretation of said text. The only way to do this is to provide quotations which allow the reader to understand where the essayist is coming from.
If you are writing a literary analysis essay about the specific literary device choices of the author, then you would need quotes to even present your examples, and then you would need your own words to analyze those examples. For example, if you are going to analyze the author's use of a particular word choice/diction used, then you need to provide the context and use the quote. It would make an essay stronger to quote the use of color imagery that provides context than to merely mention the color by itself.
All the answers given above are extremely helpful. I would add that quotations can add to the interest of an essay by giving us a very exact sense of what the short-story writer actually wrote and how, precisely, the writer wrote what s/he did. In this sense, quotations are much preferable to paraphrasing. Judicious use of quotations can also add, quite simply, to the literary interest of any essay about short fiction. When quotations are used we are, in a sense, getting "two writers for the price of one": the writer of the essay and the writer of the story. In addition, there is some phrasing in some short stories that is much more effective when quoted that when paraphrased. For instance, the grandmother in Flannery O'Connor's short story "A Good Man Is Hard to Find" emerges from a car wreck and memorably says, "I believe I have injured an organ." Quoting her exact words would be much more effective that writing something like this: "The grandmother states that she thinks some part of her body has been hurt."
Both previous posts cover pretty much every reason for quotes in essays. I'll just add this: we can see the influence of college-level essays in news stories across the world. When we read a story, we take it for granted that the writer has done the research and taken the time to verify his facts. When quotes are used in the body of the text, it adds another layer of verisimilitude ("the appearance of, or resembling, truth") to the story; we see a person's actual words and believe that the writer is fairly analyzing the context. In the same sense, when we write an essay, we want the grader to believe that we know what we are talking about, and applying knowledge of the text -- in quotes to separate it from our opinion -- gives both context to the argument and another layer of verisimilitude.
The primary reason for using quotations in essays about short stories or any other literary work is to provide support for your argument by providing textual evidence for your ideas and assertions.
Your argument is the thesis statement you present (thesis is a shortened form of hypothesis) about what you will examine, analyze and prove.
Your thesis is the statement of your argument, which may be a claim or question about a piece of literature. Your argument is something with which other people might disagree. Therefore it requires proofs. These proofs come in the form of critical opinion, literary background--and--quotations from the text.
For instance, if your argument is that Finney is wrong to suggest in "Contents of the Dead Man's Pocket" that marital entertainment time is more important than time spent successfully running a business--an argument many people would disagree with--you will present a stronger and more convincing case by using quotations from the story that show the mistake of Finney's thematic assertion. Your quotations will prove your argument and provide textual evidence for your thoughts, ideas, assertions, and claims.
"You won't mind though, will you, when the money comes rolling in and I'm known as the Boy Wizard of Wholesale Groceries?" (Finney, "Contents of the Dead Man's Pockets")
Your question is a little confusing as worded. Do you mean to say, Why are quotations necessary in literary essays about a short story?
I will just answer the general question of whether or not quotations are necessary in literary essays.
I would say they are not absolutely necessary, but usually preferable. A teacher may require them, but that is different from saying they are necessary. It is certainly possible to write a good literary essay without quotations.
Whether or not quotations are necessary, or preferable, largely depends on what you are examining. If you are looking at plot or dramatic structure or theme, for example, there may be no good need to include quotations. On the other hand, if you are examining dialogue or imagery, it would be difficult to write a good essay without quotations.