For a couple reasons, I’m sure. Here’s one of mine: to turn it around and let them see it.
“Here are four proofs written by your classmates. Which of them is the best? Why? Which of them comes in second place? What would the second place proof need to change in order to tie for first?”
Such good conversations arise when students explore decent examples of their own work compared to their classmates. And they don’t have to be time-consuming. If the work suits it, you could create a 5-minute opener comparing just two pieces of work. It can be a wonderful way for a student to recognize his/her own mistakes without me, as teacher, having to reveal them. Such recognition is a wonderful evidence of internalization of the content… real learning that can be used to solve problems.
Why do you collect student work?