First of all, let's define what an argumentative essay really is. An argumentative essay is a type of essay where you take a position on a subject, and argue that your stance on that chosen issue is correct. You use evidence to help support your argument. Typically, you will write a standard five paragraph essay with an introduction, three body paragraphs, and a conclusion. Your thesis will be the basis of your argumentative essay. It will go in your introductory paragraph. I will give you an example based on "The Most Dangerous Game," as stated in your question.
In "The Most Dangerous Game," the main character, Rainsford, must fight for his life in a human hunt with General Zaroff. If you are assigned an argumentative essay pertaining to this story, you would want to look for something within the story that is controversial. Luckily, this story has a lot to choose from. Let us use the subject of big game hunting, and its moralistic implications. You should choose a position on this subject and proceed to prove that what you are saying is right. Example: In "The Most Dangerous Game," by Richard Connell, hunting for the sake of sport, or trophy killing, is wrong on all levels.
With the example thesis above, you would then choose three strong examples from the story that prove that big game hunting is immoral. Choose a quote from the story that matches each of your three points. This backs up what you are arguing for or against. It is very important, unless you are instructed to do otherwise, to write your essay in the third person, academic voice.