To like (romantically or otherwise), understand, or approve of someone. A: "OK, you understand everything I told you?" B: "Yeah, I dig you man." I'm going to ask Cynthia out on a date tonight, I really dig her.
See also: dig
1. tv. & in. to understand something. I just don’t dig what you are saying.
2. tv. to appreciate something; to like something. He really digs classical music.