How does this person-sending-us-a-check scam work?

I will point out that although this does smell like a scam and walk like a scam, until and unless they ask for some of the funds back, it might not be a scam.

You could take the check to your bank, tell them the story, and ask what to do. If they say “It’s good, deposit it!” then fine.

I am a expert on bank fraud, and yes this smells, but listen to the actual bankers advice.

Likely they will say it is a scam, but they have the info we don’t.