Foucault's essay is an implicit response to Barthes's famous essay "The Death of the Author." The object of the essay is a problematization of authorial identity in terms of post-structuralist textual and discursive politics.
The eclipsing of the subject of discourse by the order of discourse is the post-structuralist position that problematizes the author-god of modernity in terms of textuality. The text writes itself as the author stands castrated, de-activated. Foucault thinks, author is just a way of preventing the deconstructivist infinite proliferation of meaning and is thus is related to the fear with which one wants to stop meaning from becoming infinite.
He further distinguishes between the writer of a book and the author of a book. The writer is a more unproblematical identity than the latter as the author is more of a voice within the textual discourse and gets subsumed by the discourse, as the example from Beckett Foucault cites shows. He also says that a writer only writes a book but that does not make him or her the author of discourse. Mrs. Radcliffe wrote The Mysteries of Udolpho, but that does not make him the author of the Gothic novel discourse, at large.
This is the thesis (for what it is worth) - look at the explanation below the quote and read the study guide links. Critical theory is difficult and I find Foucault one of the most confusing. Reading Roland Barthes may help you to understand Foucault. I hope this helps.
"It is obviously insufficient to repeat empty slogans: the author has disappeared; God and man dies a common death. Rather, we should reexamine the empty space left by the author's disappearance; we should attentively observe, along its gaps and fault lines, its new demarcations, and the reapportionment of this void; we should await the fluid functions released by this disappearance," (121).
Foucault wants to discuss the relationship between an author and a text, and the manner in which the text points to the author as a figure who is outside the text, and who precedes the text (and creates it). Eventually, Foucault will talk about the author as a Derridean "center" of the text, the place which originates the text yet remains outside it. (Then, of course, he will "deconstruct" that center/author).
In dealing with the "author" as a function of discourse, we must consider the characteristics of a discourse that support this use and determine its differences from other discourses. If we limit our remarks only to those books or texts with authors, we can isolate four different features.
First, they are objects of appropriation; the form of property they have become is of a particular type whose legal codification was accomplished some years ago.
Secondly, the "author-function" is not universal or constant in all discourse. Even within our civilization, the same types of texts have not always required authors; there was a time when those texts which we now call "literary" (stories, folk tales, epics and tragedies) were accepted, circulated and valorized without any questions about the identity of their author.
At the same time, however, "literary" discourse was acceptable only if it carried an author's name; every text of poetry or fiction was obliged to state its author and the date, place, and circumstance of its writing.
The third point concerning this "author-function" is that it is not formed spontaneously through the simple attribution of a discourse to an individual.