Your Apple Watch needs newer versions of iOS what does mean

According to the specs included, it looks like your iPhone 13 is still running on iOS 15.

You haven't mentioned what model of watch you're trying to pair with the phone, but I'm guessing that it is a newer model which requires a phone to be running iOS 16.

Go into the settings of your iPhone. Click on General, > Software Update. There should be an update available for it. Do the update and then try again.

Newer Apple Watches require higher iOS versions to pair with your iPhone. The newest ones require iOS 16, and it seems you are on iOS 15. See Update your iPhone or iPad - Apple Support