This issue can occur when your Bluetooth headphones have hit the limit for the number of connected Bluetooth devices. All Bluetooth devices have the capability of remembering 2-8 devices. Once that device gets close to that capacity, it will start to act oddly OR refuse to pair to new devices. A reset or factory reset should resolve this issue.
If this doesn't solve the issue, check that all devices are fully charged and reset both the AirFly and Headphones/Speaker. To reset the AirFly Duo/Pro just turn the device on and hold down the button on the side for 4 seconds. Then, reset your headphones according to their product manual instructions. Here is a brief set of instructions for resetting your AirPods. https://support.apple.com/en-us/HT209463 On Step 4 instead of connecting to the iPhone, connect to the AirFly.
Please make sure all other Bluetooth devices are off or have Bluetooth turned off. Then put the AirFly into pairing mode. After that, put your Bluetooth device into pairing mode, and it should now connect to the AirFly.