File entities and file/image fields do not currently work with the new REST GET/POST/PATCH/PUT operations.
The plan is to include file contents in base64-encoded form when serializing file entities (e.g. for GET requests). File/image field serialization should behave like for entity reference fields in general, providing a URL for the referenced file. Upon deserializing (POST requests), the content will be decoded and put in a new file on the server.
Read about how to POST aka create entities https://www.drupal.org/node/2098511
More documentation about REST: https://www.drupal.org/documentation/modules/rest