There are two ways to turn off your AirPods reading your messages from your iPhone:
If you never want to use the Announce Messages function, you can remove it from the Control Center. Go to Settings > Control Center and find Announce Messages. Tap the – (minus) next to the Announce Messages.
Once you remove it from the Control Center, Siri will not announce any incoming messages. The only way to use Announce Messages with Siri is to re-enable it again using the same steps.