How do you explain the "祖" in "祖国"?
祖 generally means ancestor, 祖国 means someone's own country and MAY have the implication that his/her ancestors lived in the same country as well, although it doesn't always have to be the case. I think 祖国 is similar to "motherland" in English, an expression to imply close relation. As an analogy, although "mother" and "land" give you a feeling what the word "motherland" means, being really serious about the meaning of "mother" itself won't help much here.

P.S. As a Chinese, I usually perceive Chinese words as a whole rather than separating every character and studying them, but I'm not sure if this is the best way for learning Chinese as a foreign language.