How can I upgrade iPhoto to be compatible with macOS Catalina?

You cannot run iPhoto in Catalina. But you can use your iPhoto Library in Photos, iPhoto's replacement. Launch Photos with the option key held down. From the resulting dialog, choose your iPhoto Library. Photos will create a new Photos Library from your iPhoto media. Your iPhoto Library will not be changed.

When you open your iPhoto library in Catalina, it will update for you.