How do I update YouTube with my iPhone?

Your iPhone 5s is too old for the current YouTube app.

You will have to use the YouTube website pending acquiring a newer (used, refurb) or newer iPhone.

You can certainly submit feedback about this app requirement (to Google YouTube app support), but I’d doubt they’ll be extending their app support as far back as iOS 10.

If you are actually running an old iOS version, the message may also mean that you cannot get the latest Youtube update until you have updated your iOS - which may not be possible if you have old hardware.

That is a message from YouTube, go to the App Store app and see if there is a YouTube update available.