I have the exact same problem.
Tested the screen on another mac (where display driver have not been installed on before) and it worked fine after the drivers were installed.
Tried reinstalling the drivers on my on mac, no luck.
The LED for mine turns on as well but only "standby LED".
So I am following this thread as well, hoping someone could help. It seems like we are not the only ones. So far, of newer threads, I found this one and another one regarding the this issue. So together with my thread it brings it to a total of three threads with the problem so far on 5 different macs running the same version of OS X. Could there be an issue with the Displaylink software and OS X 10.8.4?
Link to my thread (with more data on my system, log and so on):
http://www.displaylink.org/forum/showthread.php?p=69997
Other thread on the same issue i mentioned (he has the problem on three macs + another guy in the thread with one mac):
http://www.displaylink.org/forum/showthread.php?p=69998
*EDIT:
Just remembered that the other macbook I tested my screen on had 10.8.4 installed (and was actually totally identical), so maybe it is not 10.8.4 after all?