The Vault acts as a filter on the child's demo account or bank account. It will still be the same money in the account, but The Vault makes it clearer how much the child can buy for and how much they want to save.
To transfer the money, you click on The Vault inside the child's phone. In there (see image below) you can either enter the amount to be saved or move the slider to the correct amount. When you click save, the money can be seen divided into the account and The Vault on the start page.