In general "cleared" means gone, not dormant.
HPV is more easily passed on to another person when there are visible warts present and for at least three months afterwards.
Clearing of virus is likely related to strain and to biology/genetics of the individual person. Apparently, if the infection clears there is little (researchers rarely like to say "no") risk of cervical cancer.
Although most women will probably clear most HPV infections, with repeated exposure the odds of a nonclearing infection will increase. Since the link between cervical cancer and HPV infection is strong, and since the vaccination can provide immunity to most infections it will prevent a significant number of cervical cancer cases.
"In 2006, an estimated 10,000 women in the United States will be diagnosed with this type of cancer and nearly 4,000 will die from it. Cervical cancer strikes nearly half a million women each year worldwide, claiming a quarter of a million lives."
There's a lot of good info here (the source of that quote).
BTW, I have a 10 yo daughter and will likely get her vaccinated at age 12.