I hope you have installed android studio, SDKs and Tools for your platform as specified on React native ‘get started’ guide.
Now start your app in debugging mode. It should run you app in the android emulator that you are running. Open android studio. Select the ‘Profile or Debug APK’ option from right side key bob menu. Select any APK. You will get the interface of the android studio with project loaded. (I don’t know any other way to get to the Network profiler.). Now simply select the ‘Profiler’ tab at the bottom and click on ‘Plus’ icon on the left to attach to the application running in emulator.
Now you can see all the network calls of the application.