This is part of the general Deconstructive/Post-structuralist project: that is to decenter the source of meaning, to show the multiplicities of meaning and interpretation. For Barthes (or Derrida), there is no source: there's only destination.
When Barthes says the "birth of the reader must be at the cost of the death of the Author," he means that the meaning of a literary text (or any writing really) is not limited to the Author's intent. Previous historical understanding of an Author are: 1) the Author as a communicator, a poet, or a shaman, one who is a conduit of some Truth from God or Nature, and 2) the Author gives birth to a text, creates it; thus, all understanding of a text ultimately leads back to the Author, the Creator of the text and therefore, its meaning.
Notice that Author is capitalized in this essay and "reader" is not. This is because previous conceptions of the Author are similar to God in creating. If the Author has that kind of "author-ity" over a text, the meaning of the text is only of His/Her doing. Writing is the medium and the reader is just the passive receptacle for drinking in information the way a computer is programmed to perform functions.
Barthes sees the author in a different light. For Barthes, the Author (and the modern New Critic who is more like a shaman trying to understand that Ultimate, Transcendental Signified Meaning of the Author/God); the Author is not the origin of the text's meaning. Barthes replaces the word Author with scriptor. The scriptor does not precede the text as an all-powerful Author/Mother/God-Creator would. The scriptor is simultaneous with the text as if he/she were a character him/herself. The scriptor, with respect to a text, is more like an interpreter of history and culture, or like a chemist who takes chemicals (analogously, ideas) and mixes them together: the result is the text/mix. Then he shows what he's mixed to the reader. What is the point of the mixture(text)? The point(s) or meaning(s) are not limited to, or originating from the Author(scriptor). The reader, the destination, is where the multiplicity (freedom) of a text's meaning is manifested. The reader(s) has the power to read the text in various ways; to limit the meaning to just one Original Creator(Author) is not really a slight on God; rather, the limiting of that meaning would be like saying, 'as soon as the text is created, it ceases to grow because the meaning is 'fixed' and is only attributable to the Creator." - As if the Author were a parent, the text his child and the Author spent his entire life being the Only influence on the text/child, therefore making it impossible for the child to grow or interact with others (readers).
But for Barthes, the scriptor is not like a parent and the text is not like a child. Thus the meaning(s) of a text is not limited to either. The child is the reader. "The birth of the reader must come at the death of the Author." In a twisted but logical sense, the reader gives birth to a text by reading it. The reader liberates it (and him/herself) by not limiting it to some Original meaning. This is part of a general philosophy that a reader or student is not some passive vessel waiting to be programmed but an active participant in the construction of meaning and history.