Oracle有一个有用的教程Creating a GUI With Swing。跳过使用NetBeans IDE学习Swing部分。
下面是修改后的GUI：

创建一个JPanel放在JFrame中有助于分离您的关注点，并一次专注于GUI的一个部分。
使用JFrame比扩展JFrame更好。不应该扩展任何Java类，除非您打算重写一个或多个类方法。
Java数组必须足够长，以容纳数组的所有元素。
下面是完整的可运行代码。

import java.awt.BorderLayout;
import java.awt.FlowLayout;
import java.awt.event.ActionEvent;
import java.awt.event.ActionListener;

import javax.swing.BorderFactory;
import javax.swing.JButton;
import javax.swing.JFrame;
import javax.swing.JPanel;
import javax.swing.JTextField;
import javax.swing.SwingUtilities;

public class JTextfieldExample implements ActionListener {

    public static void main(String[] args) {
        SwingUtilities.invokeLater(new Runnable() {
            @Override
            public void run() {
                new JTextfieldExample();
            }
        });
    }

    JTextField textField1;
    JTextField textField2;
    JTextField textField3;
    JButton button;
    int[] newArr = new int[3];
    int num1, num2, num3;
    // we need to declare button and text field outside constructor so they
    // become global

    JTextfieldExample() {
        JFrame frame = new JFrame();
        frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);

        frame.add(createMainPanel(), BorderLayout.CENTER);

        frame.pack();
        frame.setLocationByPlatform(true);
        frame.setVisible(true);
    }

    private JPanel createMainPanel() {
        JPanel panel = new JPanel(new FlowLayout());
        panel.setBorder(BorderFactory.createEmptyBorder(5, 5, 5, 5));

        // Text field 1
        textField1 = new JTextField(4);
        panel.add(textField1);
        // Text field 2
        textField2 = new JTextField(4);
        panel.add(textField2);
        // Text field 3
        textField3 = new JTextField(4);
        panel.add(textField3);
        // i Button 1
        button = new JButton("Submit Guess");
        button.addActionListener(this);
        panel.add(button);

        return panel;
    }

    @Override
    public void actionPerformed(ActionEvent e) {
        if (e.getSource() == button)// if source of event comes from button...
        {
            newArr[0] = Integer.parseInt(textField1.getText());
            newArr[1] = Integer.parseInt(textField2.getText());
            newArr[2] = Integer.parseInt(textField3.getText());
            System.out.println(newArr[0] + newArr[1] + newArr[2]);

        }

    }

}