Apple Watch app is at the moment not standalone. It therefore still requires you to bring your phone with you to record your training. The watch app however make it possible for you to start/stop your recording and to see live measurements on your wrist while training.
Edit training/Pick horse
If you want to add a note, horse or ground to a training you need to open it from the app and then press edit in the training view
To get the Equilab apple watch app to work you need to have the app installed on your phone, be logged in and then open the app from the watch.
Keep the app alive during the whole training
App staying on foreground can be set on the Watch app settings from the phone. Go to: General > Wake screen > Apps > Equilab and selecting Custom > After pressing digital crown