Because that's how people talk.
Really. Show me a character who speaks entirely in grammatically-correct sentences and I'll show you awkward, unrealistic dialogue (or my cousin Henry). Real humans use local jargon and casual speech--so, of course, if you're going for realistic characters they should too.
"...some juniors like Julius, some seniors like me and Ariel."
"...some juniors like Julius, some seniors like Ariel and I."
The first one sounds more fluid, more like what I would actually say--and therefore much more natural to me. While the second is "correct," I would choose the first, especially when used in dialogue.
"My ears still ring, either from the file or the ancient copy machine, so I waste no time putting the green car into reverse and stalling (twice) before maneuvering the car onto the main road where my poor driving skills are free to terrorize the not-so-general public, meaning most people are inside, and I encounter few obstacles on the way home."
What's wrong here? It's a run-on. However, in the story, it works, because the main character isn't thinking so clearly and her mind is racing too much for shorter sentences. All of this seems to be happening at once to her, so the run-on works, even though run-ons are a formal no-no.
One trick to see how natural your dialogue is: read it out loud. If you want to test the flow of something, see how it sounds when you read it. Is it something you'd actually say? Try getting someone else to read it too--they don't know how it's supposed to sound, so you can hear how your words sound to someone else.
Any other favorite rules to break?