They don't by themselves, ever
Some apps can pull new versions using an internal updater, still requires you to install it manually.
Most apps do not have this, and its the application itself, not the APK.
APKs themselves can not auto update, they always need an updater. Either manually downloading the new APK, the application pulling the new APK themselves (for you to manually install), or an app repo client (such as the play store, f-droid, etc.)