I have an iPhone 6, which says it is currently running the most up-to-date ios (12.5.1). But when I try to download an app, it says I need 13.0.

An iPhone 6 cannot update past iOS 12.5.1. If you need iOS 13 or newer, you will need a new phone.

The app is not compatible with your phone.

You will need a newer phone if you want to use it.

I'm sorry to tell you that your phone is now too old for some apps. Newer apps use features that are only available in iOS 13 or 14 and your phone does not support them.