As many others may be discovering today, the Free App of the Day is gone and all free apps are moved to "Amazon Underground". According to Amazon, these are special editions of apps that don't even include in-app purchases. Their statement says they worked out an arrangement with developers to pay them per-minute the app is used and Amazon picks up that charge.This is a separate section on Amazon.com and requires a separate Android app.
link: http://www.amazon.com/b/ref=nav_shopall_adr_banjo?ie=UTF8&no...
More information on how they made the apps free: https://www.amazon.com/gp/feature.html?ie=UTF8&docId=1003016...
Edit: to add more information as discovered.