This phrase means to be alone. However, "alone" sometimes sounds sad and lonely:
It sucks being alone on Valentine's Day.
"On my own" sounds like a happy adventure:
My wife is taking the kids to her sister's place, so I'm on my own this weekend!
Being "by yourself" falls somewhere in the middle, neither happy nor sad:
A: Is Maxine with you?
B: No, I'm here by myself.