There's a new Chromecast update (v1.8.22) making its way out to users today that finally enables custom backdrops. You should get the option to enable this feature upon opening the updated app. If you don't feel like waiting, we've got the APK below.
Backdrop support allows you to customize the content that shows up on the Chromecast-connected screen when it's not in use. The standard feed of images is nice and all, but now you can choose your own pictures, news, weather, and even satellite images. You should be asked if you want to enable backdrops as soon as you open the updated app.
You get to opt into Chromecast backdrop customization for each device, and your personal photos are off by default. The new Backdrop section of the app also includes links to the articles (in Newsstand) from which the images came. The featured images pulled from Google+ will open in full screen so you can download them or set as wallpaper on your device. It's nifty.
The APK is signed by Google and upgrades your existing app. The cryptographic signature guarantees that the file is safe to install and was not tampered with in any way. Rather than wait for Google to push this download to your devices, which can take days, download and install it just like any other APK.
File name: com.google.android.apps.chromecast.app-1.8.22-1008022-minAPI9.apk.
Version: 1.8.22(Android 2.3+).
[Thanks, everyone who sent this in]