Normally, the boot process on a computer is displayed on all connected displays. This unfortunately doesn't seem to necessary hold on all modern Apple laptops.
There's a workaround you could try though:
- Connect an external display, keyboard, and mouse (ideally, ensure beforehand that this setup is working),
- shut down the system (if it's not turned off already),
- open the broken display, then
- start it up. The Mac will now start booting and of course try to display all information on the broken display.
- If you now close the lid during boot, the system will force the video signal to the (functioning) external display. And having an external keyboard (and mouse), you should be able to use your computer!
In case you have another Mac at your disposal, you could try and hook them up together and start the one with the broken display in "targetdisk mode". (Look up this keyword if the solution above doesn't work and you need more information as to how to hook two Macs up with one another and/or how to start a Mac in target disk mode.)