When I open the Stack Exchange beta app for iOS it opens, shows the blue background and then just flashes out and the home screen will open again.
What is wrong with the app? Is there an outage?
This is most likely related to an expired Testflight build. There is no indicator anywhere obvious that this is causing the problem, but it most likely is. (You can see that the build is expired in your Testflight app.)
The only option out is to download the official app from the App Store, or Stack Overflow Inc to update the build in Testflight.