How to recover files from a phone with a broken/unresponsive screen running Android Marshmallow?
Turn off phone using power button.
Boot into recovery mode. 
Connect it to the PC and most probably it would be connected as MTP. If not, then recover your files via adb. 

Edit: As pointed by @Firelord this will work on custom recoveries only. 
The other method is as follows. 
If you're connected to a Wi-Fi network, then find the IP address of your phone, most likely from the router setup page. You can then use ssh to copy files from the device. If ssh is not an option, then boot into fastboot mode. Flash a custom recovery for your phone and then follow the original answer.