How do I access a UI element from another controller class in JavaFX?

When you load the FXML for the secondary controller, do this:

FXMLLoader fxmlLoader = new FXMLLoader();
fxmlLoader.setLocation(getClass().getResource("second.fxml"));
AnchorPane frame = fxmlLoader.load();
FXMLSecondaryController c = (FXMLSecondaryController) fxmlLoader.getController();

Then you can pass references to the second controller. This could just be the TextArea.

EDIT:

Replaced the load() call in the snippet above and added the setLocation() call. The old line AnchorPane frame = fxmlLoader.load(getClass().getResource("second.fxml")); was wrong as it called the static load function, which is useless here.

EDIT:

(Changed the code snippet above to better match your variable names.)
The code snippet above replaces this part of your code:

AnchorPane frame = FXMLLoader.load(getClass().getResource("second.fxml"));

That line uses the FXMLLoader to load the view and also creates the instance of the controller – in this case FXMLSecondaryController. However, you cannot use the static FXMLLoader.load method for that, you need an instance of FXMLLoader. That instance holds a reference to the controller after load and you can retrieve that with getController(). And you need to cast the returned value to your controller class (with (FXMLSecondaryController)).

Your primary controller has a field:

@FXML private TextArea myArea;

This holds a reference to your TextArea and is initialized by the fxml loader. (If you remove the @FXML annotation, the loader won’t touch it.)

In your secondary controller, add this field:

public TextArea primaryTextArea;

Note that @FXML is gone (the fxml loader shall not touch the field) as well as private (you want to set the field, hence others must see it).
Now you can set this field after the controller has been loaded. So back to the loading code:

FXMLLoader fxmlLoader = new FXMLLoader();
fxmlLoader.setLocation(getClass().getResource("second.fxml"));
AnchorPane frame = fxmlLoader.load();
FXMLSecondaryController c = (FXMLSecondaryController) fxmlLoader.getController();
// Add this too:
c.primaryTextArea = myArea;

EDIT: Replaced the load() call in the snippet above and added the setLocation() call. The old line AnchorPane frame = fxmlLoader.load(getClass().getResource("second.fxml")); was wrong as it called the static load function, which is useless here.

The FXMLSecondaryController now has a reference to the TextArea of the primary controller. In one of its methods you should be able to access its content:

public class FXMLSecondaryController implements Initializable {
    // ...

    public TextArea primaryTextArea;

    // ...

    @FXML
    private void doSomething(ActionEvent event) throws Exception {
        primaryTextArea.appendText("Hi ho!");
    }

}

Note that this solution is not the best approach, but it is simple and can serve as a start. Using binding is certainly recommended. I would try to create a class which holds the data and on which the other controllers bind. But if you take the simple approach for now, it should be sufficient.