After you have logined in FreeTextSMS.net/web web application, scroll left column to the bottom, you can see a 2D bar code.
Click "LOG IN" button on the Android app, then scan this 2D bar code, this will pair your Android phone with your web browser.
After all are OK, you can leave the Android app to background and no need to check it anymore. All you should do is to enjoy your SMS messages operation on your web browser. Even you reboot your phone in future, the app will run itself and pair to web browser automatically.