Much of what makes horror effective is the use of anticipation. Horror works by persuading us to dread what is to come.
By creating not just a horror story about discovering another plane of existence but an allegory of the fall into knowledge of good and evil, Lovecraft enhances our sense that the new knowledge we are about to encounter is evil. This plays into our preexisting cultural indoctrination that knowledge often has fearful consequences that we cannot anticipate.
Elements of allegory include Tillinghast's house being "set back from Benevolent Street." This helps us anticipate that whatever Tillinghast has discovered, it is not good and will not bring benefits.
As Tillinghast understands, the narrator, who is an allegory for the reader, will want to know what it is that Tillinghast has discovered that has so altered him. Tillinghast plays on his (and our) "growing curiosity and fascination," just as Satan does in Eden.
The creatures the narrator first encounters as he gains access to the larger reality are an allegory for our misunderstandings of the true magnitude of evil. The narrator thinks these "great inky, jellyish monstrosities" are the evil he fears, only to find out the real evil is far worse. This works as an allegory for Adam and Eve's blindness about what they were unleashing when they ate from the Tree of Knowledge of Good and Evil.
The story can also be seen as allegory of the Frankenstein/Promethean myth, which shows that some knowledge is so dangerous it should be left alone. Like Victor Frankenstein, the negative physical effects of his research leave Tillinghast's body altered in terrible ways, revealing the strain of what he has encountered. This frame of reference, too, builds horror by encouraging us to anticipate the worst.