It sounds like you accidentally have the output of the Cubase mixer routed back into the input of the channel you're recording..? When you record a track, you should not get the previous track[s] mixed in with that signal. This would normally not happen if you're monitoring in headphones, but it could happen if the audio were going through speakers in the same room as the microphone -- are you sure the speakers are turned off?
It that's not it, then check that the input selected for the track being recorded is just the microphone you're using to record. If that's also set up properly, then I can only guess that maybe there's some routing in the software that's causing the main stereo output of the Cubase mixer to be routed back into the input of the track you're recording. That wouldn't normally be the way signals would be routed in a DAW, but I suppose it could happen, especially if the UR 22 comes with its own little utility software. Since I don't have either AI 8 or a UR 22 here, I can't check into it further, but that's what you'll want to look into:
- verify that the input to the recording channel in AI 8 is only getting signal from the microphone
- verify that there are no speakers outputting sound in the room -- sound is only playing through the headphones while you're recording
- verify that there's no internal routing from the main stereo output back to the currently recording channel input within AI 8 (check all input & output settings] or via any software utility that might have come with the interface