Usually this is because of a damaged or malfunctioning display but rather than write a lengthy response, I'm just going to point you to the MacBook Air Black Screen troubleshooting page.
Your inability to output to an external monitor is generally related to FileVault being enabled. If your disk is encrypted, it will often not output to a second monitor until the disk is unlocked. If your keyboard lights don't come on, this could your issue. To do that, you need to log in, a tricky endeavor with no display. Try one of the following.
- Turn your Mac off (press and hold power for like 15 seconds). Turn it on and wait a reasonable amount of time for it to boot up. Press the first letter of your username and then hit return. Type your password and return again. Hopefully you get video out at this point.
- Turn your Mac off. Then close the lid most of the way. Be sure your external monitor is connected. Press the trackpad to initiate power on and then close the display as quickly as possible. If it doesn't power on try again, but open the display a little further. If you're lucky, the timing will shove the output to an external display.
- Disconnect the internal display from the logic board, forcing the use of the external monitor.