For the answers that have been featured, do the Socratic "staff" do it, or is there anything where people can like click?
Each subject has a reviewer.
Featured Answers are selected by reviewers--the Socratic team, or "staff", is not involved in the process.
The team decided to give the more expericend members of the community the chance to review and select answers that showcase what our community can do.
So, each subject gets its own featured answers reviewer, but we do have some reviewers that cover multiple subjects.
You can find the full list of reviewers in this post.
More info on featured answers here.