Pratchett's Hogfather has an interesting take on that.
Spoilers...
The original Tooth Fairy (the whole thing is now a franchise started by him) is a bogeyman who realised milk teeth, being parts of the body, could be used to control the people they belonged to through sympathetic magic and, lacking a proper occupation due to people no longer believing in bogeymen, decided to collect them and keep them safe to prevent anyone from controlling their previous owners.
Also tooth fairies tend to carry a pair of pliers, in case they're short on change and need to take an extra tooth.