The game data is stored on your device. You will need a program that allows you to access this data. I can not recommend any such software personally, however once you have this the game data can be found
/Documents/games/com.mojang/minecraftWorlds/<WorldName>
You should be able to use the same software to download the necessary files, and then transfer them to your new phone.
You should also be able to use iTunes to back up your old phone, and then restore it to a new phone, even if they have different basebands but provided they are compatible.
As a side note, if you just want to keep your phone, its easy to have the screens fixed/replace nowadays. And generally a cheaper option if your device is not under warranty.
This question has been asked and answered before, but I could not find the previous question thread, so I went ahead and answered it again. If you find the thread please don't downvote me, I searched for it, and even checked my history as I was the one who answered it before as well.
Quitting to the title automatically saves your world on every platform that Minecraft currently exists on except for console. Your world should be fine after you quit to the title.
In the lite version of Minecraft PE it's impossible to save worlds. That can be quite confusing when migrating to the full version.
In short just Quit to Title and your world should be saved.
Hope this helped.
Best Answer
Minecraft PE does not save information on servers, it saves it on the device's local storage. If he was playing on an Android with removable storage, the transfer is as simple as using the old card in the new device.
If the old broken device can still open Minecraft then you can transfer the world by hosting a LAN game from it, joining it with the new device and hitting
Quit and Copy Worldin the Pause menu.If not, you'll have to transfer the data across using a USB cable via a computer. If the old device is broken to the point where this won't work, there's not much you can do I'm afraid.