Montresor, assuming himself wronged and (worse!) insulted by Fortunato, decides to avenge his good name. About revenge, he says: "I must not only punish but punish with impunity. A wrong is unredressed when retribution overtakes its redresser. It is equally unredressed when the avenger fails to make himself felt as such to him who has done the wrong."
This means that the nature of revenge, in his opinion, must mean that he must do so in such a way that he himself will never be suspected or caught; there can be no risk to his own person or reputation during or after the act. The reason for this is that the wrong is not really "redressed"--properly avenged--if "retribution overtakes its redresser," meaning that he has made no progress in "winning" his revenge if he has to pay for it, legally or socially. Hence the need for covertness.
Further, the wrong cannot be considered properly avenged unless the target of his vengeance, Fortunato, is not fully aware of what is happening and who is doing it. Fortunato must, in the end, be fully cognizant that Montresor has avenged Fortunato's "insults" to his person.