I don't need you to worry for me 'cause I'm alright...

They will tell you, you can't sleep alone in a strange place
Then they'll tell you, you can't sleep with somebody else
Ah, but sooner or later you sleep in your own space
Either way it's okay, you wake up with yourself

I don't need you to worry for me cause I'm alright
I don't want you to tell me it's time to come home
I don't care what you say anymore, this is my life
Go ahead with your own life, and leave me alone

I think what Billy Joel was trying to get across is that you have to live your life for yourself and not for other people. Whatever I do, I'm the only one I have to wake up with in the morning, not anybody else. I want to find my own space, where I'm ok with the me.

I am at home with the me. I am rooted in the me that is on this adventure.