Broken MacBook Pro screen
Unfortunately your question provides no details in terms of the model of MacBook Pro you have or what type of other Mac you may have access to. This is important because the easiest ways to transfer data involves hardware - eg. the ports available, whether the hard drive is easily removable, etc.

Below are some options for you. Which ones are valid will depend on your hardware.

External monitor

Plug your MacBook Pro into an external monitor so you can access/use it. This will give you a whole heap of options, such as:


Connecting an external hard drive to transfer data across
Use file sharing to make the MacBook Pro accessible to other computers
Set up your computer to use iCloud to sync your data to another Mac
Making a Time Machine backup and using this to transfer data
Use a USB flash drive to transfer data


Another point to consider with this approach is that you could still keep using the MacBook Pro by connecting it to a monitor.

Use Target Disk Mode

Start up your MacBook Pro in target disk mode using these steps: 


Ensure the MacBook Pro is off.
Connect the MacBook Pro to the other Mac using a FireWire or Thunderbolt cable. 
Power up the MacBook Pro while holding down the T key. 
Since your MacBook's screen isn't working, you won't see when it's ready - so just wait until you see the MacBook Pro's hard drive appear on the desktop of the other Mac.


Now you can access the MacBook Pro's data just like it was on any external hard drive.

Remove the hard drive

Depending on your MacBook Pro model, removing the hard drive may be a fairly easy process. If this is the case for you, then remove the hard drive and place it inside a hard drive enclosure (they're not expensive). This allows you to connect it to another Mac as an external drive. Not only can you transfer your data, but you can keep using the drive as an external drive for more data storage or backup purposes.