I need to get some more experience with writing grant proposals, and I know that professors are often invited to review them. How can I get involved in it while still being just a graduate student?
Reviewing grant proposals as a student is tricky, at least in some fields. The NSF is highly unlikely to ask a grad student to review proposals (it might theoretically be possible for a brilliant student who is almost done with their Ph.D., but I've never heard of it happening). Furthermore, faculty are not allowed to show proposals they have been asked to review to their students (it may sometimes happen, but it's breaking explicit rules regarding confidentiality). Overall, in pure math grad students basically never review grant proposals. I can't speak for other fields, but I'm skeptical that grad students ever play a major role in reviewing proposals.
Instead, I'd recommend asking your advisor to see the other side of the process. They could share their own proposals, and perhaps even reviews of those proposals or drafts of upcoming proposals. They could also ask collaborators whether they had any proposals they would be willing to share. This isn't quite the same as reviewing proposals yourself, but it could still give you valuable experience with how grant applications work.