I agree with the taxes thing. The adults pay taxes to the government, the government funds the Pokemon centers. Kind of how you pay taxes in the real world and they go to pay for running schools and stuff.
@Slipstream the trainer's license makes sense, but I don't think that your money would go to that. It could be like when you get a driver's license, you have to pay.. and that money goes to the government from DMV (or what ever it is in other countries). I don't think it would be right to make money away from someone that young like that. Maybe when you get to a certain age, you have to do stuff like that. I don't think we will ever find out unfortunately.