If I'm not wrong, 자면서 means "While sleeping"
자다 - to sleep
-면서 - while
이야기 means "story", so 이야기하다 means "to tell a story" or sometimes "to talk" or "to speak"
-고 있어요 means that it is in a present progressive tense, which means it is happening right now.
So together, they mean "while he/she/they are sleeping, I/he/she/etc am/is/etc is telling a story."