Just install the app on the new device. Make sure that you are logged into the same Apple ID on AppStore on the new device. After installing the app, just open the app. After the initial slides of introduction and the app will ask for permissions. You can allow or deny permissions, the app should show home screen directly, restoring subscription (if active) automatically.
Just in case subscription does not get restored automatically when you see a screen to subscribe to a plan, tap the Restore button to restore your existing valid subscription.
Successful restoration will automatically restore your TextApp number and you will be able to use uninterrupted services.