The quote in question appears about halfway through the short story "The Tell-Tale Heart" by Edgar Allan Poe. To understand its significance, it is first important to learn some background. The story is narrated by a person who insists over and over that he is not mad. However, it is quickly evident that he is mad, because his justification for sanity is how cleverly he can plan and execute a murder of an innocent victim. As he confesses, his only motive for killing the person in whose home he resides is that he doesn't like the look of one of the old man's eyes.

Regardless of the weak motive, once the narrator makes up his mind to kill the old man, he intently focuses on the task. He deceives the old man by acting kindly towards him. At the same time, though, he stalks him each night by opening the old man's door and carefully putting his head and a darkened lantern into the room. This process takes an hour, according to the narrator, as he does it so slowly and meticulously. When he is in position, he lets a single tiny ray of light from the lantern fall on the old man's eye, which is always closed. We can see by these intense and meticulous preparations the narrator's cunning and resolve.

On the eighth night that he does this, the narrator makes a slight noise with the lantern, and the old man becomes suspicious. The narrator remains still and silent, and he senses that the old man is terrified. The narrator imagines all sorts of excuses for the noise with which the old man must be consoling himself, and he says,

Yes, he had been trying to comfort himself with these suppositions: but he had found all in vain. All in vain; because Death, in approaching him had stalked with his black shadow before him, and enveloped the victim.

Now we are in a position to understand this quote. The word "Death" is capitalized as if it is a person, and in this context, the narrator is the personification of death. The narrator, "Death," is stalking the old man. "His black shadow before him" may refer to the dark preparations that the narrator has been making before killing the old man. The narrator has "enveloped," or surrounded, "the victim," the old man, because his evil preparations have been so thorough that he has left the old man no escape. Additionally, the "black shadow"—or, as Poe calls it in the next sentence, the "unperceived shadow"—represents the old man's dreadful premonition of his own death.