How to update iMessage apps

These apps are updated just as iOS apps are. Note in your screen shot you have a "7" showing in the App Store meaning you have seven apps to update. Go to the regular iOS App Store and update them.

Use iMessage apps on your iPhone, iPad, and iPod touch - Apple Support

When you open the App Store App and click on updates are there still apps waiting for update?

That notification number is telling you how many apps you have ready to be updated.

Go to App Store app on your phone. Tap updates on the bottom. Update All and those apps will update.

And are you signed in properly at Settings > iTunes & App Stores? Have you changed countries or regions?

Are you connected to Wifi? Did you see the apps updating? Reboot your phone. Press and hold BOTH the power and volume down buttons at the same time (hold past the swipe to shutdown) until the Apple Logo appears, then let go and your phone will restart. See if the app number goes away when your phone comes back up.

OK....Force close the Message App and App Store App and then see what you see when you open them again.

I see. That takes you to the App Store. I guess it's just saying there are 8 new things you could look at, but apparently, there is nothing for you to update there. You can't actually update anything in the App Store. You can go there to shop. Since there's actually nothing for you to update, if that number bothers you, go to Settings > Notifications > and turn off Bade App Icon for notifications or notifications altogether for App Store. The number should disappear.

I'm confused, are the apps still on your App Update page in the App Store? Are the apps still functioning?

Are all apps on your iPhone associated with exactly one Apple ID (the same Apple ID for each and every app, namely, your Apple ID)? They should be.