Thread:Meganstay/@comment-26569650-20150703170429/@comment-24577221-20150703182304

The short answer is that you need to "bind"/connect your game to an E-mail address, Facebook account, or Game Center. Then if you start MSM on your new phone and log in in the same way as before, all of your stuff will be there -- the server keeps track of your game and sends all of the necessary info to your new phone.

The more detailed explanation is in the MSM FAQs -- look here.