In order to extract the data from a smartphone, you need to go into the phone and enable file transfers. Unfortunately, you need a working screen for this. In a pinch, you can try:
# Replace the screen with a new one and perform the data transfer
# Download the files from cloud storage (hopefully you had data sync turned on)
# Download a free file recovery software (like “Recuva” or “Ibeesoft") to your computer, attach your phone via USB, and try to have the recovery software search your phone for files to transfer, if it can.
# Contact a professional data recovery service like Secure Data Recovery or Gillware.