Why do toothbrushes have to be dried?
The good people at the American Academy of Pediatric Dentistry have a number of useful resources available to shed light on questions such as yours.

I direct you to a manuscript entitled Microbial contamination of toothbrushes and their decontamination in which it is written:


  Storage conditions of toothbrushes are an important factor
  for bacterial survival. Dayoub et al. and Meier et al. reported
  that the number of microorganisms in the toothbrushes kept
  in aerated conditions was lower than in toothbrushes stored in plastic bags. Several authors have reported that bacterial contamination can be reduced by washing toothbrushes after use, and drying in aerated conditions. Caudry et al. reported that a wet environment increases bacterial growth and cross contamination. Therefore, as time increases between one toothbrushing and another, more microorganism development can occur in the toothbrushes a wet/moisture environment.


The references cited in the above passage are detailed in the linked PDF. Happy brushing!