You have found a girl (or a guy) who is worth holding on to - "a keeper."
If you go fishing, and you catch a small fish, you throw it back. But if you catch a big one, you keep it. It's a keeper. So you can extend the metaphor - you are "fishing" for a good mate, and you find one that is nice (educated, has a job, is rich, whatever it is you are looking for), so you keep her, in the sense that you don't keep looking for someone better.