What does continue running background apps mean?

Running in the background also allows any Chrome extensions you might have installed to continue to operate. These are applications that run within Chrome and either enhance the browser’s existing features, or add new ones.

Why Chrome is always running in background?

This might be to enable functionality such as email checks or virus scans, or because an app needs to update itself or stay aware while you work on other tasks. Google Chrome for Mac runs installed extensions and Web apps in the background, if they request it.