People say this a lot and I believe it's based on this quote:

I pop the video in, and wow… Tears welling, silence, goose-bumps… Wow. [I felt like] I just lost my girlfriend, because that song isn’t mine anymore… It really made me think about how powerful music is as a medium and art form. I wrote some words and music in my bedroom as a way of staying sane, about a bleak and desperate place I was in, totally isolated and alone. [Somehow] that winds up reinterpreted by a music legend from a radically different era/genre and still retains sincerity and meaning — different, but every bit as pure.

Many people seem to interpret "that song isn't mine anymore" to mean "that song is now Cash's". But here's another quote:

Then I got a CD in the post. I listened to it and it was very strange. It was this other person inhabiting my most personal song. I’d known where I was when I wrote it. I know what I was thinking about. I know how I felt. Hearing it was like someone kissing your girlfriend. It felt invasive.

To me, that doesn't sound like he's saying it's Cash's song now or that Cash's version is better. It's more that Cash proved to him that even though this was a deeply personal song about his own life and struggles that he's not the only one capable of doing it justice. He realized that other talented artists can take that song about him and transform it into a song about them. The song's not his anymore because it could be anyone's song.