The app probably gets it's content from an internet site, and is unable to connect to that site. This could be caused by AirPlane mode being on, as it turns off all radios, but that is unlikely. I see you have an wi-fi signal indication in the status bar, and AirPlane mode would disable that. So the warning is mostly just a bad guess on the apps' part on why it can't get to the necessary content.
If you can get on the internet in Safari, the problem probably has to do with connecting to the specific site. It could be blocked (for some reason) on the network you are using (unlikely), or it is down at the app's site. Or something else.
I'd go to the Apple Store page and visit the developer's home and or support pages, and see if they are reporting any problems or asking for assistance.
That's all I can think of.
Good luck.