The apps are stored locally on your computer. Depending on your operating system, here are the paths:
Mac: /Macintosh HD/Users/Applications/AirspaceApps
The apps will only appear in the folders if you have successfully downloaded and installed them. You can run the apps manually without going through Airspace Home by going to the folders mentioned above.
Airspace Home logs in to automatically check and see if there are any new purchased apps to download or if there are any updates for the apps you own.
Generally, when the app icons don't load up in Airspace Home, it means that it's connection is blocked, usually by a firewall. Do you happen to have any type of firewall installed? Windows and Mac OS come with a built-in firewall. Try disabling it and see if that resolves your issue.
Let me know if you continue to experience an issue.