Some people will claim it means you need to literally kill off a character (and all our characters are our darlings).
I'm in the camp that it means sometimes you have to kill a scene, sentence, or section you've slaved over and are attached to, but it doesn't serve the story.
Should the darling be killed? Things to consider when considering murder:
Is it in character? Does the section or scene elaborate on the character? Is the person acting in character, and if not is there a reason. Character transformation sometimes means the character does act differently, but in evaluating the darling, the new behavior should be a natural outcome of the previous action.
Are you overwriting or overexplaining? If you've shown the character, setting, or plot in other areas, you may not need to restate it.
As always, how you choose to implement this suggestion comes down to being objective about your own writing. Sometimes this is difficult. If your darling is one you feel strongly about, give it some time, a few days to a month or so and then go back and review the scene in question. If you feel it serves the story, then leave it, if not, take it out.
I'd love to hear your thoughts. Have you ever had to kill a darling? What was it, how did you feel afterward? Comment here or email me at firstname.lastname@example.org