Never heard of it until I saw your post and can't use it outside of US anyway.
Only thing I'd check is where you installed the app fro - their support says that the app store version does not support Airplay, only the version downloadable from their site. You've got to love the way Apple stupidly limits app functionality. (Having said that I've not found any apps that can Airplay other than iTunes (until Mountain Lion comes out potentially anyway). Maybe the Mog player just uses the iTunes apis and it's radiostream engine so Airplay becomes available to the outside app.
Apple are tightening up on software distribution/installation - as you probably know the only official way to get applications onto iOS devices like iPhones is via the app store and itunes - you can't just download from somewhere and install. In some ways this is good and improves security, but bad in that you're tied to their distribution system and they then control what you can use on your computer.
OS X is being made more iOS like and while the app store is actually very handy especially when you have several Macs, Apple are very strict about apps submitted for sale (or for free) in terms of things they can and can't do on the system - consequently several apps don't work in exactly the same manner as they would purchased direct from the software developer and installed manually. It's often wise to check with the software vendor - many will move to app store only distribution - those who need more low level system access (or who produce apps Apple wouldn't allow) will likely continue to offer these for sale/download. Main problem is that buying on the app store does not give you a licence to use the version from the vendor's own site more often than not, though some seem to allow some kind of migration/upgrade path if needed.
Of course Apple will maintain that it's all to protect end users etc, but don't forget they get a nice cut from app store sales whereas they get nothing if you buy from the vendor direct.
I do like the app store but I do hope they don't eventually shut out alternative modes of software installation. Mountain Lion already has a 'Gatekeeper' system that will limit how you can install software - this will be adjustable supposedly between app store only, digitally signed software with certificates from 'trusted developers' and unsigned software from any developer. I think it's a good idea for the average user to reduce the chance of installing something malicious, but not for those who want more control over their machine it might prove limiting if the options to install anything are removed.