I, like Lisabet, first assumed that this was supposed to be a post about doing damage to your characters. I wrote an admirably long post on it, but realized it sounded like a lecture in my undergrad narrative class and that if I posted it, you'd hate me.
So I've decided to look up the origin of the phrase - it came from William Faulkner - and indeed it's about cutting your purple prose down to size. This is ironic, coming from Faulkner, who didn't always take his own advice.
The theory is that writers fall in love with certain clever lines, wonderfully written bits of description, ironic asides. Basically, if it makes you proud you wrote it, you're supposed to eye it with suspicion and, if at all possible, get rid of it.
It's fair to say this point of view on writing comes out of the 'good, plainly told story' world of the early twentieth century, and probably is better advice for journalists than novelists, especially literary ones. I get the impression that if Cormac McCarthy ever took this to heart, he would never have written Suttree or Blood Meridian - two of my favourite books.
So, I'm torn on the issue. Of course you should write a story that is intelligible and cut out all the confusing but clever stuff that doesn't contribute to it. On the other hand, the experience of reading fiction is more than just the gleaning of a plotline. Mood, depth of character, nuance, voice and sub-text all contribute to the reading experience. If we all just stuck to the story arc, we'd be writing the same five stories over and over again. After all, Twilight really is Cinderella.
Nonetheless, I've had a recent experience co-writing a piece that not only made me work 500 words harder than I've ever written them before, but approximates the 'Kill your Darlings' advice when it comes to fiction, especially erotic fiction, in two ways.
Often, what I take away from a memorable erotic story are extremely vivid, filmic stills of certain erotic scenes. Of course the plot and the characters count, but what lingers for me, erotically, are these snapshots of intensely hot moments. So, of course, I try to write them figuring that, if they get me hot, they'll work for my readers.
In fact, one of the reasons I'm such an atrocious plotter, is because I really do envision my fiction, before I get down to writing it, as a series of these snapshots.
Unfortunately, I often get stuck on them. My wonderful, sublimely hot filmic still is like a bone to my pitbull mind. I can't let go of it - willfully blind to the fact that it doesn't fit the characters, or the plot, or the moment, and expend a lot of wasted energy trying to make it fit. These are darlings I have to learn to kill.
The other way in which I think that the murder of your most beloveds has relevance is in dialogue. And I've noticed I'm not the only writer with this particular flaw.
It's the placing of the perfect phrase in your character's mouth; well, it certainly seems like the perfect one at the time. You close your eyes and imagine your character saying it and, damned if it doesn't send chills up your spine. It's just so 'right'.
Except that you realize after a while that it's so 'wrong'. It simply isn't the best way your character could relay the information, it overwhelms the progression of the discussion, and you try like hell to make it fit because - oh, god - it's such a good line. 'It's just a dumb line of dialogue! Kill it!' This is a mantra I must learn and a discipline I need to practice more often.
I can't claim to be very good at killing my darlings dispassionately. I realize I should commit wholesale slaughter far more often than I do. Personally, I rely on dear friends to tell me when it's time to start the culling.
However, it's important to remember that the "Kill your darlings" advice originates from a journalistic style of writing, where personality, narrative tone, etc. need to be kept to a minimum. And I've read many a good first draft of erotica that has been castrated into toneless, mirthless reportage by too much darling murder, and ends up flat and boring.
I say murder, but in moderation.