"on my own" has a couple different meanings. It means that are living alone and supporting yourself without help. I also suggests you don't have the support of someone, usually someone you love like a boyfriend/girlfriend or your parents or that your family and friends aren't nearby. For example, "My boyfriend left me and now I'm on my own." or "I moved to a different city for a job and I'm on my own here."

"on my own" = alone