Stilted prose can kill a manuscript. You can have a great concept and nuanced characters, but if the writing is stilted, the impact of the story can be lost. Agents reject manuscripts for stilted dialogue or stilted writing in general, and so will editors.
So what is it, exactly, and how can you fix it?
Stilted writing is writing that, for one reason or another, sounds forced or unnatural. It's a wording issue, so the bad news is if your MS has it, it likely shows up on every page. The good news, though, is that it normally doesn't require replotting or shifting character arcs. It's an issue you can work through steadily and make measurable progress, page by page.
One of the major offenders in stilted prose is lack of contractions. People almost always think and speak in contractions, so when we read them on the page, they sound unnatural. "I could not get there in time" sounds much less smooth than "I couldn't get there in time." Writers sometimes don't use contractions because they want to make the sentence more formal, but there are ways to do that without sacrificing natural-sounding prose.
Another major thing that makes writing sound stilted is unnecessarily formal wording. This happens a lot in science fiction and fantasy genres, because a common way to portray cultural differences is formality. Aside from this being a really narrow scope for showing cultural differences, overly formal phrasing usually doesn't accomplish showing the dignity that writers are going for. Characters (and the narrator) can come across as stuffy, voiceless, and wooden.
A final element that will almost always make your prose sound stilted is having your characters think and speak unnaturally. Sometimes we get so used to hearing over-dramatic, unrealistic dialogue in the media around us that it filters into our writing. Stop to make sure that your dialogue and your characters' thoughts sound like people in real life, and not someone else's projection. Another element to check for is that the words you're using aren't ones that people have to use a dictionary for or are simply just words people almost never use. Readers may know what "fractious" and "nonsensical" mean, but they still stand out on the page. We don't use those words in normal conversation, so your characters shouldn't, either. (Of course, there are exceptions. But the fact that they're exceptions means that most stories don't fall in that category.) Finally, avoid having your characters call each other by name too often. Particularly when there are only two people talking, there's almost no reason for them to use the other person's name. Listen to conversations people have around you. Unless I'm yelling down the hall for him, I almost never use my husband's name when I'm speaking to him. The same goes for my family and friends. Because of that, it stands out in manuscripts as unnatural. So, double-check your conversation scenes to make sure that when it does happen, it's intentional and necessary.
Stilted prose is a bigger issue than just these things, but if you work through each of them, your voice and prose will be livelier and more natural.