Gee, no spoiler alert? I am reading "Nihil Unbound" at present, slowly, and about two-thirds done, so not really qualified to answer. It does need context in relation to Badiou and Meillassoux, with whom I'm scarcely familiar, and it certainly can get baffling.
The "death of the subject" is, of course, an old theme in the Continental tradition, and the pronouns lead me to assume Brassier is referring to such. First, it refers not to the "topic" but to whatever transcendental identity observes and constructs the "object" or "nature." The active, subjective equivalent of "substance," we might say. And known colloquially as the "human mind."
This unifying "subject" has presumably been "dying" ever since Hume's analysis of the "self," eroded and displaced by Copernicus, Marx, Darwin, etc., up to Derrida and company. At the same time, a strange inversion has taken place beginning with Kant. If there is no knowable "mind-independent" reality, then all of reality must in some way be indexed to mind, thus placing this disintegrating "subject" alone at the center of the universe. A void surrounded by an anthropocentric bubble. Philosophy as a big selfie stick. This is the delusion the "new realists," post-humanists, and radical ecologists want to reject.
Referring to Wilfred Sellars, Brassier points out the radical incommensurability of the phenomenal image of "man in the world" and the scientific image of man as a self-hallucinating cloud of neurons, atoms, cells, etc. As far as I get it, he sees the latter and the expansion of science as our "true being" yet one that steadily annihilates what we have called "meaning" or "humanity."
This is way too simplified. But he is attempting some unusual epistemological and logical contortions in the effort to arrive at a mind-independent foundation that is not a return to some pre-Kantian "naive" realism.
As to the quote.That our understanding of "ourselves" as coherent beings is "already dead" could be taken as a simple technological reality. We literally "live" inside a scientific-technical-mathematical support structure whose operating "truth" is an inhuman universe utterly indifferent to "human" existence and "already" in that sense mind-independent. The "view from nowhere" that science it criticized for holding is in fact the reality or "nowhere" we are creating.
It is an interesting radicalization of science. I'm also, personally, reading a bit of Marx into it. It reminds me of the way "living labor," including mental labor, transforms itself into the "dead capital" that replaces it. That's my quick take so far, but I'll have to grind on.