It's "I am not my own, for i have been made new".
I am not on my own = I am not alone / I am not by myself.
But:
I am not my own = I don't belong to myself (implying that I belong to someone else, in this case God.)
For I have been made anew = I am a new person (changed or saved by God)
The lyrics of the song come from the Corinthians, in the Bible, so it's a Christian worship song I think.