In your User model, add the following line in the User class:
public $timestamps = true;
Now, whenever you save or update a user, Laravel will automatically update the created_at and updated_at fields.
Update:
If you want to set the created at manually you should use the date format Y-m-d H:i:s. The problem is that the format you have used is not the same as Laravel uses for the created_at field.
Update: Nov 2018 Laravel 5.6
"message": "Access level to App\\Note::$timestamps must be public",
Make sure you have the proper access level as well. Laravel 5.6 is public.