To put the mic into pairing mode, hold down on the power button on the mic until the LED turns blue. This may take up to 20 seconds. While holding down the power button, the LED will go from white, to off, to white again, and finally blue after about 20 seconds.
You then need to "forget" the Audigo mic in your iPhone's Bluetooth settings before restarting the app and pairing the mic again to a previously paired device.
To pair the mic, tap on the small mic icon in the upper right of the app or the large icon in the middle of the record screen.