No, Airhome does not require an internet connection, but it does require a Wi-Fi network.
You can listen to all of your downloaded music and movies locally on Airhome without an Internet connection using AirPlay on your Wi-Fi network. AirPlay works with iPhone, iPad, and iPod touch with iOS 4.3.3 or later, Mac with OS X Mountain Lion or later, Mac and PC with iTunes 10.2.2 or later. You can learn more about how to AirPlay in the "How to AirPlay" Category.
For the best Airhome audio and video Wi-Fi entertainment experience, it is recommended that you use broadband high speed internet service with the minimum connection speeds listed below for streaming audio and video content from the internet:
An internet connection is required to access online streaming services for music and movies, including internet radio stations and Apple TV.