One of a handful of reasons... Here's the list, in order of simplicity (easiest to most difficult or expensive):
1. I'm working on the assumption that anything is possible, and that you are amateur in repairs. First check to make sure you're using the appropriate display assembly. This model can use both the 2010 and the 2011 11.6" display assembly with the 30-pin (I believe it's 30... I'll check) LVDS cable connector. While both a 2012 & 2013 looks very similar, they are not compatible because the LVDS cable is ever so slightly different. Make sure you have the right one. That being said...
2. The LVDS cable isn't seated properly. Reseat all cables (this procedure should be done while the battery is disconnected... ALWAYS)
3. The pins on the new display's LVDS cable (or on the port of the logic board that receives it) are damaged and you will either need a new display (if the LVDS cable coming from the display is damaged), or need a new logic board (or repair the current one) if the problem lies with the receiving port. The only real sure fire way to test this is to use a known good working display and swap it in. Does the known good display work?
IF YES, the replacement display you have is faulty in some way (check the LVDS cable pins and the backlight ribbon connector underneath the hinge. I can take a pic for you if you end up needing to do this.
IF NO then the logic board was damaged somehow and it will need repair or replacement. If you don't know how to do that yourself (which I'm guessing you don't), then many of us on here can recommend good solid vendors who repair these boards without spending a fortune. Usually, a rework like that is <$200 for the consumer.
4. The backlight on the display is either faulty (either it came that way, or you shorted it somehow -- from improper ESD handling protocol or having the battery connected while swapping the screens) or the ribbon cable that connects from the actual backlight LED strip to the small circuit board underneath the hing of the display that powers it has come loose. It isn't likely, but it has happened to me before with refurbished display assemblies. This is a bit harder to get to, but easily remedied if you know where it is and how to get to it.
That's about all I can think of... try to get ahold of a known good working display (ie, one you physically see light up in front of you... take it off that macbook air, and try it with yours. Make sure you're pulling a 2010/2011 display module).