I can't grasp from reading the description of this extension what its purpose is. What can I do once I install this extension?
Purpose of this API is unclear
It's an extremely simplistic proof of concept. You give it a text and it returns the same text with misspelled words wrapped in <span class='misspelled'>. You need a PHP-compatible spelling engine for the content language installed on the server.
Could this at some point become a VisualEditor plugin? Should it be added to Mentorship_programs/Possible_projects?
Yes. In fact, I already have a draft for such a project and I'll add it soon.