My phone's screen is broken and is useless, but it has important files on it. Is there a way to back the files up or get the phone back?
You can only back data up yourself, if the phone is functioning (apart from the screen obviously). Make sure it shows some kind of life signal when you plug it in/turn it on/press the volume keys/etc. It could be something like the charging LED lighting up or vibration. If you have the SIM card in there, you could also call it and check that way. If none of these work, then the only solution is to take it to a data recovery center (it can cost a couple hundred dollars, so make sure those photos are irreplacable).
So, first of all, a broken screen could mean a couple of things:
1.: The digitizer (touch screen glass) is broken
This is the easiest problem to solve. If you have an OTG cable (a cable with a micro USB end and a female USB on the other), you can connect a mouse to the phone and use that to back everything up. It's very rare nowadays for a phone (especially phablets) to not support this feature.
You could also just buy a new screen and either apply it yourself or take it to a repair shop if you want to keep the phone.
2.: The LCD (the display) doesn't work
If you see a black screen or random lines on the screen, when you turn it on, then the LCD is broken. This has a bunch of different types too.
If you're lucky and the phone goes into file transfer mode, when plugged into a computer, then you can just take the files off, so make sure to try this first.
a.: Water damage
If you dropped your phone in water and it got damaged, then either a chip got fried or something got corroded. It's usually impossible to tell which one it is, so the phone has to be disassembled and cleaned. It's not really worth it to buy a screen at this point, LCD don't really mind water that much. However, If you're "lucky" and only the backlight is faulty (or the connectors got corroded), then you can just take a strong flashlight, hold it up against the screen (possibly somewhere with a white/light area of pixels). Make sure to try this out, in my experience, even some repair shops forget to check it. I've had it on a friend's phone recently and it looked like this:
If it's not the backlight, then probably some video output chip got damaged. Backing up data in this case is very hard. You could try using one of the tools designed for this but it's a gamble whether they work and even if they do, they might require you to turn on USB debugging (which is hard without seeing anything). I also don't like to install random programs on my computer.
In some cases, removing the corrosion from the device can also solve some problems. If you want to keep the warranty, take it to a repair shop and let them clean it. If you don't care however, iFixit has some really nice guides to disassemble almost any phone. Make sure to remove every board from the device, then take an unused tootbrush and some Isopropyl alcohol and give it a good scrub. Clean the adapters and the ends of the cables too. You're essentially looking for some blueish-greenish goo:
If you don't want to void warranty and you can't take it to a repair shop either, then you could theoretically put it in a tub of alcohol, but I haven't tried it and it certainly won't have the same effect.
After this, assemble the phone again without missing anything out and test it. Hopefully it works. If not, check the general advices part.
b.: Blunt damage
If you dropped your phone on concrete and the screen won't turn on again, it's either because the LCD got damaged or a cable got loose (the latter is very unlikely, I've only had it happen once and only with a vibration motor). If it turns on, but you see lines one the screen (see image below), then the LCD is broken.
In any case, it's usually worth it to get a new screen if it's still in production. Make sure to not touch the glass too much, the small glass shards don't mix well with your finger.
c.: General advices
If nothing above worked for you, there's a small chance going through a software based solution will work too.
If you have USB debugging enabled, install adb, pull up the command line of your choice, connect the phone to the computer through USB (pray to your choice of deity) and type "adb devices" without the quotes. If your phone show up in the list, you can continue. Type "adb pull mnt/sdcard/DCIM/Camera/" without quotes. This will pull the images from your phone. You can also do "adb pull mnt/sdcard/" if you want every file on there, because you have some sick scores in temple run you want to keep.