It lets you search based on the first 5 characters of a SHA-1 password hash. So you use a trusted tool on your own computer to hash the password you want to search, and pass in just part of it. You get multiple results, and you can then compare against the hashes yourself to see if yours is in there. There's no way for them to collect your password that way.
I suppose someone could build a bare basic UI in something like JsFiddle that is transparently auditable and has crystal clear comments for non-technical people to see what the code does
Exactly - but the reactions I saw when he first released Pwned Passwords was "this is a malicious tool, don't give your password to anyone". Even if you're hitting the API from your own service, you need the entire password first to submit the partial hash to the API.
We who understand what's going on know it's perfectly fine, but it's hard to get that message across. Just like the first time you try to explain what a hash is to a non-technical person.