My wife started having the issue with the screen going black when opening any app on Tuesday. She has an iPhone 6s that came with iOS 9. She since upgraded to iOS 9.0.1 and then 9.0.2 when it was released. The issue persisted. This morning, I performed a hard restart on her phone by holding the home and power buttons until the screen turned black and turned back on with the Apple logo. Only time will tell if that fixed the issue.
I also got an iPhone 6s at the same time. Like her, I started at iOS 9, updated to 9.0.1 on Monday, and then updated to 9.0.2 early this morning (around 3 AM EST). I have not had any issues.