Why aren't trainers allowed on golf courses?

Please can someone explain to me why trainers are a big no no on golf courses - 'proper golf shoes are required'. I never understood this and as a parent wanting to get his kids into golf, I begrudge having to pay for yet another pair of shoes every summer with the speed that they grow out of them?

I have always been told that it is 'grumble grumble GREENS grumble grumble'???