Rhost's API is HTTP based but not RESTful.
Rhost listens on a separate API port. You connect to that port, authenticating with a dbref and a password. (This password is NOT the login password but is set with @api/password obj=pw) via Basic HTTP authentication.
The dbref must have the API @power.
The API is entirely header based. If you do a GET request, the 'Exec' header evaluates code as that dbref and returns its result. If you do a POST request, the 'Exec' header is a series of commands that are run. Basically, GET is like u() and POST is like @tri.
There's some other details: POST can have a Time header which will delay the commands that long. GET can have a Parse header which will determine if it evaluates the code as normal, only as substitutions, or ansi, and some other options. It can also have an Encode header to return the content in base64.
This basically allows you to execute arbitrary code / responses in the MUSH from external scripts. And with the execscript() function you can execute external scripts from within the game.
I haven't used it a lot yet, but one way I do use it is before I would have a timer on my game to dump a flatfile every so file. Now, my backup script (run by linux's cron) uses the API to @dump/flat before it runs.