What does exacly mean by Entity CRUD API?
The CRUD API is the set of functions that allows an entity to be Created, Readed, Updated and Deleted. 

An entity has a CRUD API when its entity controller implementes de EntityAPIControllerInterface. The EntityAPIController class, the default Entity API entity controller class implements this interface.

So you only have to define your entity using the Entity API contrib module and use that class as controller (or your custom subclass of that class).

If you read the entity_export code you can see:

 if (in_array('EntityAPIControllerInterface', class_implements($info['controller class']))) {
    return entity_get_controller($entity_type)-&gt;export($entity, $prefix);
  }


The function checks if the entity controller implements EntityAPIControllerInterface interface; if yes the export method is called.