It appears to be a plea to the listener for protection, and means "don't let me hurt myself".

http://www.pinkspage.com/lyrics/dontletme.html

The expression "get + pronoun" can mean hurt, attacked, shot, killed [someone]. "I got him!" from a solider to a pal would mean he successfully shot an enemy. "The little bugger got me!" might mean a mosquito bit him. "Don't let him get me" from a person in fear of a stalker to a policeman is a plea for protection.

"Don't let me get me" is so unusual I've never heard it (why would you ask a stranger to protect you against yourself?) but it seems clear from context that that is what is meant here--consider the preceding line "I"m a hazard to myself" and the following one "I'm my own worst enemy".

*SINGING*

"...so irritating don't wanna be my friend no more-I wanna be somebody else..."

Hallo,
I think the song means that she's been making so many mistakes in her life that she should be protected from herself. The singer doesnt have good judgement and always finds herself in trouble (she slept with her professor, etc) that she thinks she needs protection from herself.
Cool song. Just voicing out insecurities most teens feel.