Welcome to iPhoneForums!
Give these a try:
Play a youtube video on your iPhone, and turn up the volume. Try recording again.
Use a third party app for recording. See if that solves your problem.
Reset all settings. This has to be done in Settings - General - Reset - Reset all Settings. You'll have to look at your settings in the Settings app afterwards, and set up your iPhone as preferred.
Make sure your iPhone is up to date. The latest software for the iPhone 5s is iOS 10.3.2.
Also make sure you're not connected to external speakers via Bluetooth when you're trying to play the videos, and that your case doesn't cover the microphone.