Why do rodents' teeth continue to grow? Enamel formation and/or root formation continue throughout rodents' lives.
What's the mechanism that causes this?
The article "Dental Anatomy of Rabbits and Rodents" (Crossley et al. 2010) provides a basic review of rodent teeth, as there are over 1700 species. But, in general, the teeth are constant replaced due to abrasion.
What the mechanism is for continual dental growth in rodents are, in part, discussed in the article "Tooth organogenesis and regeneration" (Thesleff and Tummers), where they state