Following
https://tools.ietf.org/html/rfc7231#section-6.5.13, HTTP 415 indicates to the client that it has sent a payload which cannot be processed with the requested method on the given resource, because the media type of the payload is not suitable. Like, you sent a picture when plaintext was expected.
A rest api should respond with 415 if a client makes this kind of error. The API may offer a media type with support for hyperlinks which informs the client what it is supposed to send to make life easier for the client (collection+json, uber, hal-forms, siren, hydra, hyper-json-schema or of course html), or it could indicate a profile to overlay an existing mediatype with additional semantics by sending a profile link header, where the expected mediatypes of request bodies could be part of the profile description.
Your question might also mean, how should a client handle a 415 response. Sometimes it just failed to send a Content-Type header matching the payload.
If an api uses no hyperlinks and no profile link header, your only hope is some out-of-band documentation, i.e. the api's website or a swagger page.