The short fast answer, don’t use your bluetooth headphones.
I was using the Sony MDRXB950BT/B Extra Bass Bluetooth Headphones and they were causing the error.
Update Sept 28 2016 – ScreenFlow reached out to me and let me know they can’t fix the issue because of something to do with Apple and Bluetooth.
The Screenflow Error
I had a really hard time with Screenflow just now. Here’s the error I couldn’t get rid of:
Due to an internal error, the recording has unexpectedly terminated. Please try making your recording again, and if problems persist, contact us for technical support.
Unable to start recording.
If you’re reading this and you’re with Screenflow, please let me know in the comments section if there is an update to the software which will allow me to use my headphones again.