To look for common writing problems.
When writing essays or papers, it's completely natural for us to make mistakes. That is why we proof-read out work to correct those mistakes, but even then, we may miss some things. That is why a peer review is necessary. A different set of eyes will almost definitely find the mistakes that we've missed, and will point them out to us.
Furthermore, just as a writing suggestion; if you're writing a long paper, take a break, stand up, walk around, but LEAVE the computer. When you go back and read your work, you'll see some mistakes you didn't before. Happens to me every time!
