The most basic answer is to improve your grammar and overall structure of your project. If you're writing novels, this is where you make your money and your reputation. Poor grammar = poor reviews = poor reputation. Having a copy editor can remove those hidden errors, like an apostrophe in the wrong place or an incorrect spelling, before the readers do.
You might think that you can easily edit your own work. WRONG! I know from experience how hard it is to go through your own work, which you've probably toiled over and worked hard on for months, and cut pieces out. Don't get me wrong, it's not impossible, but it is extremely difficult. Naturally, we have a connection with our work; it's like your baby. A copy editor can go through your work with a neutral pair of eyes. We might be cruel when we all but destroy parts of your work, but we are being cruel to be kind.
Whenever I edit something, I approach it from three points of view: an editor, a fellow writer, and a reader. My 'editor' eyes look at the grammar, punctuation, structure etc. and offers suggestions on how to improve sentences and descriptions. My 'fellow writer' brain analyses the plot and characters. Is the story believable? Are there any holes in the story line? Does it flow well? And so on. Finally, my 'reader' mind decides whether it is an engaging piece of work. It also works with the 'fellow writer' mind to determine what could be improved in terms of the setting and character descriptions if it's a work of fiction. Can I picture the scene clearly, or is it like looking through frosted glass?
It's not just authors who need a copy editor. University students are another group that can benefit from a copy editor, final year students in particular. I know how stressful final year can be (I'm there right now), especially when your dissertation deadline is breathing down your neck. You don't want the pinnacle of your degree, your masterpiece, to be tainted by grammar and typing errors.
We all make mistakes. It's a fact of life. However, some mistakes will cost you more dearly than others. In answer to the question at the start of this article, I would say if you're writing anything content related, you'll almost certainly need a copy editor. Go through your work yourself first by all means, but relying on your own editing is risky. Using a fresh pair of eyes can turn a 3* review into a 5* review.
Remember, copy editors are being cruel to be kind. We only want your work to be the best it can be!