Everyone knows to proofread their own work. After all, spellcheck won't catch everything. But here's another reason it's important to read things back to yourself: when you receive a proof of your work, carefully re-reading may help you reword something to provide clearer understanding.
For example, I recently received an article proof from a magazine designer. My part of the job was done -- I had researched, interviewed, written and submitted the article. Now I was proofing for typos and adding photo captions. But when I saw the finished product and re-read it aloud, I suddenly found a better way to say something I had already said. A cleaner, more succinct sentence never hurt anyone. And this was my chance to change it and make the article better.