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.