Most likely something went wrong during device pairing.
It is important that iPhone learns that the Mac is capable of acting as an external keyboard to iPhone. This happens during the Bluetooth pairing procedure. For this it is important that the Type2Phone application is running while the Mac and the iPhone discover one another.
If this goes wrong, Type2Phone may not see the iPhone or Type2Phone may connect to iPhone, but the iPhone will ignore keyboard input it receives from Type2Phone.
To fix, you need to have both devices forget the other one and then repeat the pairing procedure.
1. Undo the pairing: have both devices forget each other.
1a. In System Preferences > Bluetooth on Mac
1b. In Bluetooth Settings on iOS.
2. On the Mac, open System Preferences > Bluetooth and leave it open
3. Launch Type2Phone and keep it frontmost
4. Initiate pairing from iPhone Settings > Bluetooth.
If this still fails, repeat. This time, reboot both the Mac and the iPhone after undoing the existing pairing.