Why do rodents' teeth continue to grow?
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


  During the cap and bell stages the lateral sides of the epithelial bud start enveloping the underlying dental mesenchyme, and from this point onwards the leading edge of the epithelium is called the cervical loop. The basal epithelial cell layer of the loop bordering the dental papilla is known as the inner enamel epithelium and the part facing the dental follicle is known as the outer enamel epithelium. The core of the loop is populated by loosely arranged stellate reticulum cells and a thin layer of stratum intermedium cells facing the inner enamel epithelium This cervical loop structure is maintained in continuously growing teeth and constitutes the epithelial part of the adult stem cell niche (Harada et al., 1999; Tummers and Thesleff, 2003: Tummers and Thesleff, 2008)