To extract files from a broken MacBook Pro, you have two options: use Target Disk Mode or remove the hard drive from your MBP and access it with another Mac. You can try one or both, depending on your condition.
Solution 1: Target Disk Mode
Target Disk Mode is is a boot mode on Intel-based Macs that helps data transfer through Thunderbolt Bridge, which does not boot Mac's operating system but makes Mac's drive behaves as an external storage device on the destination Mac.
Here I'll show you how to boot your broken MBP to Target Disk Mode:
Step 1: Connect two Macs both through the Firewire or Thunderbolt cable mentioned above.
Step 2: Force your broken MacBook Pro to shut down by holding the power button, and then reboot Mac with the T key pressed.
Step 3: Then you'll see a Thunderbolt or FireWire icon on the screen, which means that the Mac is in Target Disk Mode.
Step 4: Wait until the second Mac detects your Mac and recognizes its startup disk as an external hard drive.
Step 5: Then you can go ahead to copy or transfer files from the target Mac to the connected Mac easily.
If this method fails to transfer files from the dead MBP, move to the next one straightforwardly.
Solution 2: Remove the hard drive from your broken MBP
You can either remove the hard drive from your old MBP and use it on another well-functional Mac device instead. You can use a USB dock like OWC to use the hard drive as an external hard drive.