My interpretation:
“Somebody else” does not refer to a third person, i.e. another woman the narrator might love, different from the one he’s crying for; it means that since the day he fell in love with her, he has not been able to live alone, without somebody at his side, because he has known how it used to be with someone to love (presumably her, whom he is crying for).
Then again, I’m not a native speaker, so it’s possible that “somebody else” would not be used in that sense.