Let’s start with the basics. Peer review is part of many disciplines. Giving feedback to a peer within your graduate program provides practice for peer review, a major element of many academic fields. You get a chance to practice what you’ll be doing professionally.
The larger benefits come from receiving feedback from those in the field. Usually, graduate students who are working on their projects focus on those projects. They’re trying to research the topic and develop their ideas. This route can result in a kind of intellectual myopia. You’re simply too close to your topic to understand it well.
That’s a variation of the benefit of getting any review. The key benefits of getting reviews from professors more advanced in the field is that they have perspective on that field. They may be able to give you guidance on which journals might accept your article. They may be able to give you feedback on where your article fits in the field. They may be able to give you a kind of intellectual context you lack. For example, you may find they recognize a similar argument that was originally made 20 years ago that you could benefit by integrating. They may recognize gaps in your argument that you do not know because they are more familiar with the larger field. If an earlier article related to the one you are writing received quite a bit of criticism, you may be able to benefit from reviewing those comments. Without the review process, you may not have known to look for it.
Finally, one benefit is very basic: these are the people who will evaluate your work and let you know if it’s good enough for you to receive your degree. If you can get feedback from them before you submit it, this may save you a great deal of time.