Thanks for posting in the Apple Support Community. I understand that you're trying to capture a screen recording on your iPhone but there is no sound from the video playing on the screen.
Any time you record your screen on your iPhone, the recording should capture the sound from whatever is playing. Is the volume turned up when you are listening to the playback and are you following these steps to capture your screen recording:
How to record the screen on your iPhone, iPad, or iPod touch
If you're following those steps, then I recommend to close out of the app that you were trying to capture a screen recording in, then restart the iPhone and test it out again.
Force an app to close on your iPhone, iPad, or iPod touch
Restart your iPhone, iPad, or iPod touch