What should I do if my App Store can't update apps and Messenger needs updating?

I notice you are running an iPhone 6s with iOS 11. Update your phone's iOS to have the minimum system version necessary (iOS 15.8) for running that app. This may be the last update you can ever install so don't be surprised if in a year or so you will be in a similar position with no solution.

There are different ways to ensure the highest iOS that your device will support is installed on your device. When updating, your device will update to the highest version currently available that it will support, or it will just say it is up to date (and can't go any higher) and won't install anything. If you want to run a higher system version than your current device is capable of running, you need to buy a newer device.

iPhones 6S through 7, and SE(gen. 1), will update to iOS 15.8.2

This support article has instructions about updating an iPhone or iPad wirelessly --> Update your iPhone or iPad - Apple Support Read the article for recommended criteria for being able to use wireless updating.

You can also use Mac and PC computers to update system software --> Update iOS on iPhone - Apple Support

Messenger is Google's messaging system through Facebook, and as stand-alone app. If you have no need for it, delete that app. Or, do what it says, go into the App Store app and update Messenger.