De Quincey says that murder singularly focuses the attention on the instinct for self-preservation, which destroys any sense of distinction between human beings and animals:
the primal law of self-preservation, is the same in kind, (though different in degree,) amongst all living creatures; this instinct therefore, because it annihilates all distinctions, and degrades the greatest of men to the level of “the poor beetle that we tread on,” exhibits human nature in its most abject and humiliating attitude.
During the murder itself, then, everything is unvaried darkness, and there are no distinctions between man and woman, human and devil:
the murderers are taken out of the region of human things, human purposes, human desires. They are transfigured: Lady Macbeth is “unsexed;” Macbeth has forgot that he was born of woman; both are conformed to the image of devils; and the world of devils is suddenly revealed.
But by moving from Duncan's murder to the knocking on the gate, Shakespeare moves away from this nihilistic moment to one where the human world reasserts itself. When the drunken porter hears Macduff and Lennox knock, his reactions, untinged by any knowledge of the murder, reestablish the everyday world of human interactions. This pulls the spectators out of being swept up in the annihilation of murder. The return of the ordinary world allows them to see how close to the abyss they have been:
when the deed is done, when the work of darkness is perfect, then the world of darkness passes away like a pageantry in the clouds: the knocking at the gate is heard; and it makes known audibly that the reaction has commenced: the human has made its reflux upon the fiendish; the pulses of life are beginning to beat again; and the reestablishment of the goings-on of the world in which we live, first makes us profoundly sensible of the awful parenthesis that had suspended them.
The knocking on the gate, by reasserting the familiar human world, paradoxically allows us to see how powerful the murder is. We have no vantage point on the "awful parentheses" of annihilation when we are actually in that perfect darkness. We can only recognize it once we are safely out of it. This is why De Quincey says that the knocking of the gate "reflected back upon the murder a peculiar awfulness and a depth of solemnity": it provides perspective that allows us to see how terrible a deed the murder is, a perspective not possible during the moment of annihilation itself.
De Quincey explicitly says that what we feel for Macbeth is "a sympathy of comprehension, a sympathy by which we enter into his feelings, and are made to understand them,—not a sympathy of pity or approbation." To say this is emphatically not to "admit feeling sympathy for Macbeth." De Quincey himself explains in a footnote that this is a terrible misreading:
It seems almost ludicrous to guard and explain my use of a word in a situation where it would naturally explain itself. But it has become necessary to do so, in consequence of the unscholarlike use of the word sympathy, at present so general, by which, instead of taking it in its proper sense, as the act of reproducing in our minds the feelings of another, whether for hatred, indignation, love, pity, or approbation, it is made a mere synonyme of the word pity; and hence, instead of saying “sympathy with another,” many writers adopt the monstrous barbarism of “sympathy for another.”
De Quincey says we understand what is in Macbeth's or Lady Macbeth's mind, and in that sense we are in sympathy with them. It does not follow that we feel sympathy for them in the sense of approving of their deed or even feeling sorry for them.
Obiter dicta, which fortuitously is also relevant to the discussion of sympathy: Explaining De Quincey's stance does not mean I'm in agreement with it. I don't quite buy De Quincey's reading, nor do I agree with the comment to the question that it "is one of the best critical pieces on Macbeth." It's certainly one of the most celebrated, but it tells us more about De Quincey than about Shakespeare, and it has not aged well.