Views search to find users based on terms
With search API you create an index containing all the info that should be searchable. The easiest thing would probably be to create an index based on the Profile entity and add the reference to the User entity and add the fields you need, like the term reference.

If you only want to let users search on the term reference, you might want to exclude normal search altogether and use facets instead, that would allow users to select terms and get the users displayed. Alternatively you could just use Views alone since Search API in not needed if you don't do fuldtext search.

With or without Search API you can in Views select the fields you want to display, just like any view, and use the template options Views provide to customize the output of the search page.