I am creating a new application and will have to deal with ActionBar. I know that I have to extend the AppCompatActivity because the ActionBarActivity is deprecated, however
First of all you are asking Good Question, Android will add advanced features continuously in different API levels but those features are available from which level of API they are added
For Example: Consider android fragment functionality was added in API level 11 that means it will work for API level 11 and above but your application need for API level 10 devices also at that time it wont be work. For this reason android develop support library for cover a wide range of Android devices (support for low level API) to work those functionality.
Android always recommend developers to use support library for development for more information check here