Как сохранять значение после выхода?

Можно это реализовать так:

// Java 7
private final String KEY_RATING = "KEY_RATING";
private final String RATING_PREFS_FILE = "RATING_PREFS_FILE";
@Override
protected void onCreate(Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);
    ...
    RatingBar ratingBar = (RatingBar) findViewById(R.id.myRatingBar);
    SharedPreferences prefs = getSharedPreferences(RATING_PREFS_FILE, Context.MODE_PRIVATE);
    ratingBar.setRating(prefs.getFloat(KEY_RATING, 0));
    ratingBar.setOnRatingBarChangeListener(new RatingBar.OnRatingBarChangeListener() {
        @Override
        public void onRatingChanged(RatingBar ratingBar1, float rating, boolean fromUser) {
            prefs.edit().putFloat(KEY_RATING, rating).apply();
        }
    });
}

//Java 8
private final String KEY_RATING = "KEY_RATING";
private final String RATING_PREFS_FILE = "RATING_PREFS_FILE";
@Override
protected void onCreate(Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);
    ...
    RatingBar ratingBar = (RatingBar) findViewById(R.id.myRatingBar);
    SharedPreferences prefs = getSharedPreferences(RATING_PREFS_FILE, Context.MODE_PRIVATE);
    ratingBar.setRating(prefs.getFloat(KEY_RATING, 0));
    ratingBar.setOnRatingBarChangeListener((ratingBar1, rating, fromUser) -> 
        prefs.edit().putFloat(KEY_RATING, rating).apply());
}

Также советую почитать про SharedPreferences вот тут.