I have submitted my apps to test flight and able to download it using testflightapp.com. However, when I access it through the testflight app itself, it says "You are currently not testing any app"?
I have checked the following link but it's not helpful http://help.testflightapp.com/customer/portal/articles/829653-i-don-t-see-any-apps-after-connecting-my-device-to-my-account-
I guess you donwloaded the Apple Testfligt App. It's different than the system you are using. You need to access the testflightapp-website with your Ipad/Iphone. Have a look in your emails. There should be a Testflight link. Open it with your iOS device and it should work. It will reate an icon on your home-screen which looks like an App but is just the link to testflightapp.com