You get points for suggesting edits and badges for completing review activities (votes to close, triage, etc). I thought you got points for the latter as well but looks like that's not the case.
This is a common misperception about moderation on Stack Overflow. You'll often see people claim that people get rep for doing moderation tasks. And some people do pursue badges... though the gold review badge (1000 reviews) has only been awarded 47 times on Stack Overflow ( https://meta.stackoverflow.com/help/badges/81/steward ) ... and silver (250 reviews) 65 times ( https://meta.stackoverflow.com/help/badges/78/reviewer). ... so I would find it difficult to accept that badges are things that motivate people.