Now on to the body of Joe's letter. Note that Joe is not whining. Admissions officers hate whining, and it won't get you anywhere. Joe is not saying that his rejection was unfair, nor is he insisting that the admissions office made a mistake. Instead, in both the opening and closing of the letter, he notes that he respects the decision of the admissions folks.
Most important for an appeal, Joe does have a reason to appeal. He tested poorly on the SAT, and he retook the exam and brought up his scores dramatically. Notice that Joe makes mention of being sick when he first took the SAT, but he is not using that as an excuse. An admissions officer is not going to reverse a decision simply because a student claims some kind of testing hardship. You need actual scores to show your potential, and Joe comes through with the new scores.
Also, Joe is wise to send along his most recent grade report. He is doing extremely well in school, and the admissions officers will like to see those strong grades. Joe is not slacking off senior year, and his grades are trending upward, not down.
Note that Joe's letter is brief and to the point. He's not wasting the time of the admissions officers with a long rambling letter. The college already has Joe's application, so he doesn't need to repeat that information in the appeal.
Joe's letter does three important things in a concise manner. He states his respect for the admissions decision; he presents the new information that is the basis for his appeal; and he reaffirms his interest in the college. Were he to write anything else, he would be wasting the time of the admissions officers.
Finally, it is important to be realistic about an appeal. Joe writes a good letter and has significantly better scores to report. However, he is likely to fail in his appeal. The appeal is certainly worth a try, but the majority of rejection appeals are not successful.