How can I save an activity state using the save instance state?

You need to override onSaveInstanceState(Bundle savedInstanceState) and write the application state values you want to change to the Bundle parameter like this:

@Override
public void onSaveInstanceState(Bundle savedInstanceState) {
  super.onSaveInstanceState(savedInstanceState);
  // Save UI state changes to the savedInstanceState.
  // This bundle will be passed to onCreate if the process is
  // killed and restarted.
  savedInstanceState.putBoolean("MyBoolean", true);
  savedInstanceState.putDouble("myDouble", 1.9);
  savedInstanceState.putInt("MyInt", 1);
  savedInstanceState.putString("MyString", "Welcome back to Android");
  // etc.
}

The Bundle is essentially a way of storing a NVP (“Name-Value Pair”) map, and it will get passed in to onCreate() and also onRestoreInstanceState() where you would then extract the values from activity like this:

@Override
public void onRestoreInstanceState(Bundle savedInstanceState) {
  super.onRestoreInstanceState(savedInstanceState);
  // Restore UI state from the savedInstanceState.
  // This bundle has also been passed to onCreate.
  boolean myBoolean = savedInstanceState.getBoolean("MyBoolean");
  double myDouble = savedInstanceState.getDouble("myDouble");
  int myInt = savedInstanceState.getInt("MyInt");
  String myString = savedInstanceState.getString("MyString");
}

Or from a fragment.

@Override
public void onViewStateRestored(@Nullable Bundle savedInstanceState) {
    super.onViewStateRestored(savedInstanceState);
    // Restore UI state from the savedInstanceState.
    // This bundle has also been passed to onCreate.
    boolean myBoolean = savedInstanceState.getBoolean("MyBoolean");
    double myDouble = savedInstanceState.getDouble("myDouble");
    int myInt = savedInstanceState.getInt("MyInt");
    String myString = savedInstanceState.getString("MyString");
}

You would usually use this technique to store instance values for your application (selections, unsaved text, etc.).