If the meeting is being cloud recorded, it will only record the main room, regardless of what room the meeting host is in. If local recording is being used, it will record the room the participant who is recording is in. Multiple participants can record locally.
Ask the host for permission to record. The host needs to allow recording before opening the breakout rooms. If the host has already open the breakouts rooms, they can join your breakout room to allow you to record.
Click Record in the meeting controls to start a local recording.