"A narrator should not supply interpretations of his work; otherwise he would not have written a novel, which is a machine for generating interpretations."
...is the birth of the reader.
Death of the Author is a concept from literary criticism which holds that an author's intentions and biographical facts (the author's politics, religion, etc) should hold no weight when coming to an interpretation of their writing; that is, that a writer's interpretation of his own work is no more valid than the interpretations of any of the readers.
Intentions are one thing. What was actually accomplished might be something very different.
The logic is fairly simple: Books are meant to be read, not written, and so the ways readers interpret them are more important and "real" than the ways writers write them. There are also the more practical facts that a lot of authors are not available or not willing to comment on their intentions, and even when they are, artists don't always make choices for reasons that make sense or are easily explained to others — or, in some cases, even to themselves.
Although popular amongst postmodern critics, this has some concrete modernist thinking behind it as well, on the basis that the work is all that outlives the author (hence the name) and we can only judge the work by the work itself. The author's later opinions about their work are a form of criticism and analysis themselves, and therefore are not necessarily consistent with what's written unless the author or publisher actively goes back and changes it (and even then it can be argued that, since the original work still exists, the author has merely created a different version of it). One critic's understanding of the author's background and opinions is likely to be just as accurate as another's, especially if the author has an idiosyncratic or even anachronistic perspective on their own work. Modernists are more likely to appeal to the similar but not identical concept of the Intentional Fallacy, which does not discount biographical information or other works by the same author.
Needless to say, many writers don't especially like this. Margaret Atwood famously remarked that if the Death of the Author theory became prevalent, then "we [writers] are all in trouble". However, while J. R. R. Tolkien acknowledged the influence of his experiences on his works (The Lord of the Rings), he denied that he had written allegory, insisting that his works simply had Applicability; this arguably makes him an early supporter of the Death of the Author, since pointless speculations about an author's allegorical intent are exactly what the concept seeks to avoid, in favor of analyzing the "applicability" of the text itself. It has been joked (with delicious irony) that Roland Barthes, who actually wrote the trope naming essay, probably had to say "No, that's not what I meant at all!" at least once in his lifetime while discussing it. Playwright Alan Bennett claims he responded to students asking for assistance on analyzing his works as part of their A-Levels to "treat [him] like a dead author, who [is] thus unavailable for comment".
Of course, numerous authors including the likes of Ray Bradbury and William Gibson can't be bothered to stay consistent when talking about the major themes or concepts in their books for more than a few years at a time.
Or worse, if the author comes to reject their own work, they may express dissatisfaction with certain parts and not others. Hence, "the perfect is the enemy of the good" (Translation: "coulda, woulda, shoulda"). This is why some auteur filmmakers oppose the notion of a Director's Cut on the grounds that the "real" film will always be the one people saw in cinemas in the year of release, not the ideal film in one's head.
This is a given in works where the authors don't hold copyright and can be replaced, especially Shared Universes; if a writer is fired and replaced by another, anything the old writer has stated in interviews can be (and often is) freely Jossed by the new writer.
Isaac Asimov repeated in several places an anecdote based on this: He once sat in (in the back of a large lecture hall, so semi-anonymously) on a class where the topic of discussion was one of his own works. Afterward, he went up and introduced himself to the teacher, saying that he had found the teacher's interpretation of the story interesting, though it really wasn't what he had meant at all. The teacher's response was "Just because you wrote it, what makes you think you have the slightest idea what it's about?"
There is an echo of this concept in Asimov's short-short story "The Immortal Bard", in which William Shakespeare is brought into the present day and takes a college course about his writings. He flunks.
There is an Older Than Feudalism example about some Jewish sages having an argument about their law... and ignoring God's interpretation in favor of their own. Because, you see, the Torah is not in Heaven— Umberto Eco, postscript to The Name of the Rose