Civil wars happen. All states have a civil war at one point or another on their point to modernity.
This civil war broke out quite simply because Gaddafi is unable and far too delusional to accept the will of his people for change. Rather than step down (e.g. Egypt), he decides to try to brutally crush any resistance and to maintain the status quo which suits him, not his people. Those who are mainly fighting back are defectors who have nowhere else to go. Waiting for the inevitable is just as bad, so they fight. The 'majority' you speak of are probably too afraid to be on the receiving end of a modern army when you have only rocks to defend yourself with. It's a no-brainer why there isn't swathes of people; the Libyans knew they were on their own and in a losing war of attrition. Chances are you'll see more people come out of the woodworks now the resolution has been passed and even more when the first plane gets shot down.